:: Principle of Duality in Real Projective Plane: a Proof of the Converse :: of {D}esargues' Theorem and a Proof of the Converse of {P}appus' :: licenses, or see http://www.gnu.org/licenses/gpl.html and :: http://creativecommons.org/licenses/by-sa/3.0/. environ vocabularies INCSP_1, ANPROJ11, REAL_1, XCMPLX_0, ANPROJ_1, ANPROJ_2, PENCIL_1, MCART_1, EUCLID_5, ARYTM_1, ARYTM_3, CARD_1, EUCLID, FUNCT_1, NUMBERS, PRE_TOPC, RELAT_1, SUBSET_1, SUPINF_2, ANPROJ_9, TARSKI, INCPROJ, RVSUM_1, BKMODEL1, CARD_FIL, PROJRED2, PBOOLE, RELAT_2, AFF_2, VECTSP_1, ANALOAF; notations TARSKI, SUBSET_1, XCMPLX_0, PRE_TOPC, RVSUM_1, COLLSP, INCPROJ, ANPROJ_9, XREAL_0, NUMBERS, FUNCT_1, FINSEQ_2, EUCLID, ANPROJ_1, BKMODEL1, STRUCT_0, RLVECT_1, EUCLID_5, INCSP_1, PROJRED2, ANPROJ_2; constructors MONOID_0, EUCLID_5, ANPROJ_9, BKMODEL1, EUCLID_8, PROJRED2; registrations BKMODEL3, ORDINAL1, ANPROJ_1, STRUCT_0, XREAL_0, MONOID_0, EUCLID, VALUED_0, ANPROJ_2, FUNCT_1, FINSEQ_1, XCMPLX_0, INCPROJ, PASCAL; requirements SUBSET, NUMERALS, ARITHM, BOOLE; equalities BKMODEL1, XCMPLX_0, COLLSP, INCPROJ, ANPROJ_9, EUCLID_5; expansions TARSKI, XBOOLE_0, STRUCT_0, PROJRED2; theorems EUCLID_8, EUCLID_5, ANPROJ_1, ANPROJ_2, EUCLID, XCMPLX_1, RVSUM_1, FINSEQ_1, ANPROJ_8, INCPROJ, ANPROJ_9, XBOOLE_0, COLLSP, BKMODEL1, EUCLID_4; begin ::Preliminaries theorem for a,b,c,d,e,f,g,h,i being Real holds |{ |[a,b,c]|, |[d,e,f]|, |[g,h,i]| }| = a * e * i + b * f * g + c * d * h - g * e * c - h * f * a - i * d * b proof let a,b,c,d,e,f,g,h,i be Real; reconsider p = |[a,b,c]|, q = |[d,e,f]|, r = |[g,h,i]| as Element of TOP-REAL 3; A1: p`1 = a & p`2 = b & p`3 = c & q`1 = d & q`2 = e & q`3 = f & r`1 = g & r`2 = h & r`3 = i by EUCLID_5:2; |{ |[a,b,c]|, |[d,e,f]|, |[g,h,i]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27; hence thesis by A1; end; theorem Th2: for a,b,c,d,e being Real holds |{ |[a,1,0]|, |[b,0,1]|, |[c,d,e]| }| = c - a * d - e * b proof let a,b,c,d,e be Real; reconsider p = |[a,1,0]|, q = |[b,0,1]|, r = |[c,d,e]| as Element of TOP-REAL 3; A1: p`1 = a & p`2 = 1 & p`3 = 0 & q`1 = b & q`2 = 0 & q`3 = 1 & r`1 = c & r`2 = d & r`3 = e by EUCLID_5:2; |{ |[a,1,0]|, |[b,0,1]|, |[c,d,e]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27; hence thesis by A1; end; theorem Th3: for a,b,c,d,e being Real holds |{ |[1,a,0]|, |[0,b,1]|, |[c,d,e]| }| = b * e + a * c - d proof let a,b,c,d,e be Real; reconsider p = |[1,a,0]|, q = |[0,b,1]|, r = |[c,d,e]| as Element of TOP-REAL 3; A1: p`1 = 1 & p`2 = a & p`3 = 0 & q`1 = 0 & q`2 = b & q`3 = 1 & r`1 = c & r`2 = d & r`3 = e by EUCLID_5:2; |{ |[1,a,0]|, |[0,b,1]|, |[c,d,e]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27; hence thesis by A1; end; theorem Th4: for a,b,c,d,e being Real holds |{ |[1,0,a]|, |[0,1,b]|, |[c,d,e]| }| = e - c * a - d * b proof let a,b,c,d,e be Real; reconsider p = |[1,0,a]|, q = |[0,1,b]|, r = |[c,d,e]| as Element of TOP-REAL 3; A1: p`1 = 1 & p`2 = 0 & p`3 = a & q`1 = 0 & q`2 = 1 & q`3 = b & r`1 = c & r`2 = d & r`3 = e by EUCLID_5:2; |{ |[1,0,a]|, |[0,1,b]|, |[c,d,e]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27; hence thesis by A1; end; theorem Th5: for u being Element of TOP-REAL 3 holds u is zero iff |( u, u )| = 0 proof let u be Element of TOP-REAL 3; reconsider un = u as Element of REAL 3 by EUCLID:22; hereby assume u is zero; then 0.REAL 3 = u by EUCLID:66; then |( un,un )| = 0 by EUCLID_4:17; hence |( u, u )| = 0; end; assume |( u, u )| = 0; then un = 0.REAL 3 by EUCLID_4:17; hence thesis by EUCLID:66; end; theorem for u,v,w being non zero Element of TOP-REAL 3 st |{u,v,w}| = 0 holds ex p being non zero Element of TOP-REAL 3 st |(p,u)| = 0 & |(p,v)| = 0 & |(p,w)| = 0 proof let u,v,w be non zero Element of TOP-REAL 3; assume A1: |{u,v,w}| = 0; reconsider p = |[u`1,v`1,w`1]|, q = |[u`2,v`2,w`2]|, r = |[u`3,v`3,w`3]| as Element of TOP-REAL 3; A2: p`1 = u`1 & p`2 = v`1 & p`3 = w`1 & q`1 = u`2 & q`2 = v`2 & q`3 = w`2 & r`1 = u`3 & r`2 = v`3 & r`3 = w`3 by EUCLID_5:2; A3: |{ p,q,r }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27; |{u,v,w}| = u`1 * v`2 * w`3 - u`3*v`2*w`1 - u`1*v`3*w`2 + u`2*v`3*w`1 - u`2*v`1*w`3 + u`3*v`1*w`2 by ANPROJ_8:27; then consider a,b,c be Real such that A4: a * p + b * q + c * r = 0.TOP-REAL 3 and A5: a <> 0 or b <> 0 or c <> 0 by A1,A2,A3,ANPROJ_8:42; A6: |[0,0,0]| = |[a * p`1,a * p`2,a * p`3]| + b * q + c * r by A4,EUCLID_5:4,7 .= |[a * p`1,a * p`2,a * p`3]| + |[b * q`1,b * q`2,b * q`3]| + c * r by EUCLID_5:7 .= |[a * p`1,a * p`2,a * p`3]| + |[b * q`1,b * q`2,b * q`3]| + |[c * r`1,c * r`2,c * r`3]| by EUCLID_5:7 .= |[a * p`1+b*q`1,a * p`2+b*q`2,a * p`3+b*q`3]| + |[c * r`1,c * r`2,c * r`3]| by EUCLID_5:6 .= |[a * p`1+b*q`1+c*r`1,a * p`2+b*q`2+c*r`2,a * p`3+b*q`3+c*r`3]| by EUCLID_5:6; reconsider p = |[a,b,c]| as non zero Element of TOP-REAL 3 by A5; take p; thus |(p,u)| = p`1 * u`1 + p`2 * u`2 + p`3 * u`3 by EUCLID_5:29 .= a * u`1+p`2*u`2+p`3*u`3 by EUCLID_5:2 .= a * u`1+b*u`2+p`3*u`3 by EUCLID_5:2 .= a * u`1+b*u`2+c*u`3 by EUCLID_5:2 .= 0 by A6,A2,FINSEQ_1:78; thus |(p,v)| = p`1 * v`1 + p`2 * v`2 + p`3 * v`3 by EUCLID_5:29 .= a * v`1+p`2*v`2+p`3*v`3 by EUCLID_5:2 .= a * v`1+b*v`2+p`3*v`3 by EUCLID_5:2 .= a * v`1+b*v`2+c*v`3 by EUCLID_5:2 .= 0 by A6,A2,FINSEQ_1:78; thus |(p,w)| = p`1 * w`1 + p`2 * w`2 + p`3 * w`3 by EUCLID_5:29 .= a * w`1+p`2*w`2+p`3*w`3 by EUCLID_5:2 .= a * w`1+b*w`2+p`3*w`3 by EUCLID_5:2 .= a * w`1+b*w`2+c*w`3 by EUCLID_5:2 .= 0 by A6,A2,FINSEQ_1:78; end; theorem Th7: for u,v,w being non zero Element of TOP-REAL 3 st |(u,v)| = 0 & are_Prop w,v holds |(u,w)| = 0 proof let u,v,w be non zero Element of TOP-REAL 3; assume that A1: |(u,v)| = 0 and A2: are_Prop w,v; consider a be Real such that a <> 0 and A3: w = a * v by A2,ANPROJ_1:1; reconsider un = u,vn = v as Element of REAL 3 by EUCLID:22; thus |(u,w)| = |(a * vn,un)| by A3 .= a * |(v,u)| by EUCLID_8:68 .= 0 by A1; end; theorem Th8: for a,u,v being non zero Element of TOP-REAL 3 st not are_Prop u,v & |(a,u)| = 0 & |(a,v)| = 0 holds are_Prop a,u v proof let a,u,v be non zero Element of TOP-REAL 3; assume that A1: not are_Prop u,v and A2: |(a,u)| = 0 and A3: |(a,v)| = 0; u v is non zero by A1,ANPROJ_8:51; then reconsider uv = u v as non zero Element of TOP-REAL 3; A4: a`1 * u`1 + a`2 * u`2 + a`3 * u`3 = 0 & a`1 * v`1 + a`2 * v`2 + a`3 * v`3 = 0 by A2,A3,EUCLID_5:29; per cases by EUCLID_5:3,4; suppose A5: a`1 <> 0; then A6: u`1 = -a`2/a`1 * u`2 - a`3/a`1 * u`3 & v`1 = -a`2/a`1 * v`2 - a`3/a`1 * v`3 by A4,ANPROJ_8:13; set p1 = u,p2 = v; now reconsider r = a`1 as Real; thus A7: u v = |[ 1 *( p1`2 * p2`3 - p1`3 * p2`2), a`2/a`1 *( p1`2 * p2`3 - p1`3 * p2`2), (a`3/a`1) * (- p1`3*p2`2 + p1`2*p2`3) ]| by A6 .= ( p1`2 * p2`3 - p1`3 * p2`2) * |[ 1 ,a`2/a`1, a`3/a`1 ]| by EUCLID_5:8 .= ( p1`2 * p2`3 - p1`3 * p2`2) * |[ a`1 / r, a`2 / r, a`3 / r ]| by A5,XCMPLX_1:60 .= ( p1`2 * p2`3 - p1`3 * p2`2) * ((1/a`1) * a) by EUCLID_5:7 .= (( p1`2 * p2`3 - p1`3 * p2`2) * (1/a`1)) * a by RVSUM_1:49; p1`2 * p2`3 - p1`3 * p2`2 <> 0 proof assume p1`2 * p2`3 - p1`3 * p2`2 = 0; then u v = |[0 * a`1,0 * a`2,0 * a`3]| by A7,EUCLID_5:7 .= 0.TOP-REAL 3 by EUCLID_5:4; hence thesis by A1,ANPROJ_8:51; end; hence (p1`2 * p2`3 - p1`3 * p2`2) * (1/a`1) <> 0 by A5; end; hence thesis by ANPROJ_1:1; end; suppose A8: a`2 <> 0; then A9: u`2 = -a`1/a`2 * u`1 - a`3/a`2 * u`3 & v`2 = -a`1/a`2 * v`1 - a`3/a`2 * v`3 by A4,ANPROJ_8:13; set p1 = u, p2 = v; now reconsider r = a`2 as Real; thus A10: u v = |[ (a`1/a`2) *( p1`3 * p2`1 - p1`1 * p2`3), 1 *( p1`3 * p2`1 - p1`1 * p2`3), (a`3/a`2) * ( p1`3*p2`1 - p1`1*p2`3) ]| by A9 .= (p1`3*p2`1-p1`1*p2`3) * |[a`1/a`2,1,a`3/a`2]| by EUCLID_5:8 .= (p1`3*p2`1-p1`1*p2`3) * |[a`1/r,r/r,a`3/r]| by A8,XCMPLX_1:60 .= (p1`3*p2`1-p1`1*p2`3) * ((1/a`2) * a) by EUCLID_5:7 .= ((p1`3*p2`1-p1`1*p2`3) * (1/a`2)) * a by RVSUM_1:49; p1`3*p2`1-p1`1*p2`3 <> 0 proof assume p1`3*p2`1-p1`1*p2`3 = 0; then u v = |[0 * a`1,0 * a`2,0 * a`3]| by A10,EUCLID_5:7 .= 0.TOP-REAL 3 by EUCLID_5:4; hence thesis by A1,ANPROJ_8:51; end; hence (p1`3*p2`1-p1`1*p2`3) * (1/a`2) <> 0 by A8; end; hence thesis by ANPROJ_1:1; end; suppose A11: a`3 <> 0; a`3 * u`3 + a`1 * u`1 + a`2 * u`2 = 0 & a`3 * v`3 + a`1 * v`1 + a`2 * v`2 = 0 by A4; then A12: u`3 = -a`1/a`3 * u`1 - a`2/a`3 * u`2 & v`3 = -a`1/a`3 * v`1 - a`2/a`3 * v`2 by A11,ANPROJ_8:13; set p1 = u, p2 = v; now reconsider r = a`3 as Real; thus A13: u v = |[ (a`1/a`3) * (p1`1 * p2`2 - p1`2 * p2`1), a`2/a`3 * (p1`1 * p2`2 - p1`2 * p2`1), 1 * (p1`1 * p2`2 - p1`2 * p2`1) ]| by A12 .= (p1`1*p2`2-p1`2*p2`1) * |[a`1/a`3,a`2/a`3,1]| by EUCLID_5:8 .= (p1`1*p2`2-p1`2*p2`1) * |[a`1/r,a`2/r,r/r]| by A11,XCMPLX_1:60 .= (p1`1*p2`2-p1`2*p2`1) * ((1/a`3) * a) by EUCLID_5:7 .= ((p1`1*p2`2-p1`2*p2`1) * (1/a`3)) * a by RVSUM_1:49; p1`1*p2`2-p1`2*p2`1 <> 0 proof assume p1`1*p2`2-p1`2*p2`1 = 0; then u v = |[0 * a`1,0 * a`2,0 * a`3]| by A13,EUCLID_5:7 .= 0.TOP-REAL 3 by EUCLID_5:4; hence thesis by A1,ANPROJ_8:51; end; hence (p1`1*p2`2-p1`2*p2`1) * (1/a`3) <> 0 by A11; end; hence thesis by ANPROJ_1:1; end; end; theorem Th9: for u,v being non zero Element of TOP-REAL 3 for r being Real st r <> 0 & are_Prop u,v holds are_Prop r * u,v proof let u,v be non zero Element of TOP-REAL 3; let r be Real; assume that A1: r <> 0 and A2: are_Prop u,v; consider a be Real such that A3: a <> 0 and A4: u = a * v by ANPROJ_1:1,A2; r * u = (r * a) * v by A4,RVSUM_1:49; hence thesis by A1,A3,ANPROJ_1:1; end; begin :: Alignment of definitions definition let P being Point of ProjectiveSpace TOP-REAL 3; attr P is zero_proj1 means :Def1: for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.1 = 0; end; registration cluster zero_proj1 for Point of ProjectiveSpace TOP-REAL 3; existence proof take Dir001; reconsider p = |[0,0,1]| as non zero Element of TOP-REAL 3; now let u be non zero Element of TOP-REAL 3; assume Dir001 = Dir u; then are_Prop u, p by ANPROJ_1:22; then consider a be Real such that a <> 0 and A1: u = a * p by ANPROJ_1:1; A2: |[0,0,1]| = |[p.1,p.2,p.3]| by EUCLID_8:1,def 3; thus u.1 = a * p.1 by A1,RVSUM_1:44 .= a * 0 by A2,FINSEQ_1:78 .= 0; end; hence thesis; end; end; registration cluster non zero_proj1 for Point of ProjectiveSpace TOP-REAL 3; existence proof set P = Dir100; take P; reconsider u = |[1,0,0]| as non zero Element of TOP-REAL 3; now thus P = Dir u; |[1,0,0]| = |[u.1,u.2,u.3]| by EUCLID_8:1,def 1; hence u.1 <> 0 by FINSEQ_1:78; end; hence thesis; end; end; theorem Th10: for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.1 <> 0 proof let P be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; assume A1: P = Dir u; consider u9 be non zero Element of TOP-REAL 3 such that A2: P = Dir u9 and A3: u9.1 <> 0 by Def1; are_Prop u,u9 by A1,A2,ANPROJ_1:22; then consider a be Real such that A4: a <> 0 and A5: u = a * u9 by ANPROJ_1:1; assume u.1 = 0; then a * u9.1 = 0 by A5,RVSUM_1:44; hence thesis by A3,A4; end; definition let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; func normalize_proj1(P) -> non zero Element of TOP-REAL 3 means :Def2: Dir it = P & it.1 = 1; existence proof consider u be non zero Element of TOP-REAL 3 such that A1: P = Dir u and A2: u.1 <> 0 by Def1; reconsider v = |[1, u`2/u.1,u`3/u.1]| as non zero Element of TOP-REAL 3; take v; A3: v`1 = 1 by EUCLID_5:2; u.1 * v = |[u.1 * 1, u.1 * (u`2/u.1),u.1*(u`3/u.1)]| by EUCLID_5:8 .= |[u.1, u`2, u.1*(u`3/u.1)]| by XCMPLX_1:87,A2 .= |[u`1, u`2, u`3]| by A2,XCMPLX_1:87 .= u by EUCLID_5:3; then are_Prop u,v by A2,ANPROJ_1:1; hence thesis by A1,A3,ANPROJ_1:22; end; uniqueness proof let u,v be non zero Element of TOP-REAL 3 such that A4: P = Dir u & u.1 = 1 and A5: P = Dir v & v.1 = 1; are_Prop u,v by A4,A5,ANPROJ_1:22; then consider a be Real such that a <> 0 and A6: u = a * v by ANPROJ_1:1; A7: 1 = a * v.1 by A4,A6,RVSUM_1:44 .= a by A5; a * v = |[a * v`1, a * v`2, a * v`3]| by EUCLID_5:7 .= v by A7,EUCLID_5:3; hence thesis by A6; end; end; theorem Th11: for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| proof let P be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; let u9 be non zero Element of TOP-REAL 3; assume P = Dir u9; then Dir u9 = Dir normalize_proj1 P by Def2; then are_Prop u9,normalize_proj1 P by ANPROJ_1:22; then consider a be Real such that a <> 0 and A1: normalize_proj1 P = a * u9 by ANPROJ_1:1; A2: normalize_proj1 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7; A3: 1 = (normalize_proj1 P)`1 by Def2 .= a * u9`1 by A2,EUCLID_5:2; then A4: u9`1 = 1 / a & a = 1 / u9`1 by XCMPLX_1:73; normalize_proj1 P = |[ 1,u9`2 / u9`1,(1 / u9`1) * u9`3]| by A1,A3,A4,EUCLID_5:7 .= |[ 1,u9.2 / u9.1,u9.3/u9.1]|; hence thesis; end; theorem for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 for Q being Point of ProjectiveSpace TOP-REAL 3 st Q = Dir normalize_proj1(P) holds Q is non zero_proj1 by Def2; definition let P being Point of ProjectiveSpace TOP-REAL 3; attr P is zero_proj2 means :Def3: for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.2 = 0; end; registration cluster zero_proj2 for Point of ProjectiveSpace TOP-REAL 3; existence proof take Dir001; reconsider p = |[0,0,1]| as non zero Element of TOP-REAL 3; now let u be non zero Element of TOP-REAL 3; assume Dir001 = Dir u; then are_Prop u, p by ANPROJ_1:22; then consider a be Real such that a <> 0 and A1: u = a * p by ANPROJ_1:1; A2: |[0,0,1]| = |[p.1,p.2,p.3]| by EUCLID_8:1,def 3; thus u.2 = a * p.2 by A1,RVSUM_1:44 .= a * 0 by A2,FINSEQ_1:78 .= 0; end; hence thesis; end; end; registration cluster non zero_proj2 for Point of ProjectiveSpace TOP-REAL 3; existence proof set P = Dir010; take P; reconsider u = |[0,1,0]| as non zero Element of TOP-REAL 3; now thus P = Dir u; |[0,1,0]| = |[u.1,u.2,u.3]| by EUCLID_8:1,def 2; hence u.2 <> 0 by FINSEQ_1:78; end; hence thesis; end; end; theorem Th13: for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.2 <> 0 proof let P be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; assume A1: P = Dir u; consider u9 be non zero Element of TOP-REAL 3 such that A2: P = Dir u9 and A3: u9.2 <> 0 by Def3; are_Prop u,u9 by A1,A2,ANPROJ_1:22; then consider a be Real such that A4: a <> 0 and A5: u = a * u9 by ANPROJ_1:1; assume u.2 = 0; then a * u9.2 = 0 by A5,RVSUM_1:44; hence thesis by A3,A4; end; definition let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; func normalize_proj2(P) -> non zero Element of TOP-REAL 3 means :Def4: Dir it = P & it.2 = 1; existence proof consider u be non zero Element of TOP-REAL 3 such that A1: P = Dir u and A2: u.2 <> 0 by Def3; reconsider v = |[u`1/u.2, 1,u`3/u.2]| as non zero Element of TOP-REAL 3; take v; A3: v`2 = 1 by EUCLID_5:2; u.2 * v = |[u.2 * (u`1/u.2), u.2 * 1,u.2*(u`3/u.2)]| by EUCLID_5:8 .= |[u`1, u.2, u.2*(u`3/u.2)]| by XCMPLX_1:87,A2 .= |[u`1, u`2, u`3]| by A2,XCMPLX_1:87 .= u by EUCLID_5:3; then are_Prop u,v by A2,ANPROJ_1:1; hence thesis by A1,A3,ANPROJ_1:22; end; uniqueness proof let u,v being non zero Element of TOP-REAL 3 such that A4: P = Dir u & u.2 = 1 and A5: P = Dir v & v.2 = 1; are_Prop u,v by A4,A5,ANPROJ_1:22; then consider a be Real such that a <> 0 and A6: u = a * v by ANPROJ_1:1; A7: 1 = a * v.2 by A4,A6,RVSUM_1:44 .= a by A5; a * v = |[a * v`1, a * v`2, a * v`3]| by EUCLID_5:7 .= v by A7,EUCLID_5:3; hence thesis by A6; end; end; theorem Th14: for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| proof let P be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; let u9 be non zero Element of TOP-REAL 3; assume P = Dir u9; then Dir u9 = Dir normalize_proj2 P by Def4; then are_Prop u9,normalize_proj2 P by ANPROJ_1:22; then consider a be Real such that a <> 0 and A1: normalize_proj2 P = a * u9 by ANPROJ_1:1; A2: normalize_proj2 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7; A3: 1 = (normalize_proj2 P)`2 by Def4 .= a * u9`2 by A2,EUCLID_5:2; then A4: u9`2 = 1 / a & a = 1 / u9`2 by XCMPLX_1:73; normalize_proj2 P = |[ u9`1 / u9`2,1,(1 / u9`2) * u9`3]| by A1,A3,A4,EUCLID_5:7 .= |[ u9.1 / u9.2,1,u9.3/u9.2]|; hence thesis; end; theorem for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 for Q being Point of ProjectiveSpace TOP-REAL 3 st Q = Dir normalize_proj2(P) holds Q is non zero_proj2 by Def4; definition let P being Point of ProjectiveSpace TOP-REAL 3; attr P is zero_proj3 means :Def5: for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.3 = 0; end; registration cluster zero_proj3 for Point of ProjectiveSpace TOP-REAL 3; existence proof take Dir100; reconsider p = |[1,0,0]| as non zero Element of TOP-REAL 3; now let u be non zero Element of TOP-REAL 3; assume Dir100 = Dir u; then are_Prop u, p by ANPROJ_1:22; then consider a be Real such that a <> 0 and A1: u = a * p by ANPROJ_1:1; A2: |[1,0,0]| = |[p.1,p.2,p.3]| by EUCLID_8:1,def 1; thus u.3 = a * p.3 by A1,RVSUM_1:44 .= a * 0 by A2,FINSEQ_1:78 .= 0; end; hence thesis; end; end; registration cluster non zero_proj3 for Point of ProjectiveSpace TOP-REAL 3; existence proof set P = Dir001; take P; reconsider u = |[0,0,1]| as non zero Element of TOP-REAL 3; now thus P = Dir u; |[0,0,1]| = |[u.1,u.2,u.3]| by EUCLID_8:1,def 3; hence u.3 <> 0 by FINSEQ_1:78; end; hence thesis; end; end; theorem Th16: for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds u.3 <> 0 proof let P be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; assume A1: P = Dir u; consider u9 be non zero Element of TOP-REAL 3 such that A2: P = Dir u9 and A3: u9.3 <> 0 by Def5; are_Prop u,u9 by A1,A2,ANPROJ_1:22; then consider a be Real such that A4: a <> 0 and A5: u = a * u9 by ANPROJ_1:1; assume u.3 = 0; then a * u9.3 = 0 by A5,RVSUM_1:44; hence thesis by A3,A4; end; definition let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; func normalize_proj3(P) -> non zero Element of TOP-REAL 3 means :Def6: Dir it = P & it.3 = 1; existence proof consider u be non zero Element of TOP-REAL 3 such that A1: P = Dir u and A2: u.3 <> 0 by Def5; reconsider v = |[u`1/u.3, u`2/u.3,1]| as non zero Element of TOP-REAL 3; take v; A3: v`3 = 1 by EUCLID_5:2; u.3 * v = |[u.3 * (u`1/u.3), u.3 * (u`2/u.3),u.3*1]| by EUCLID_5:8 .= |[u`1, u.3 * (u`2/u.3), u.3 ]| by XCMPLX_1:87,A2 .= |[u`1, u`2, u`3]| by A2,XCMPLX_1:87 .= u by EUCLID_5:3; then are_Prop u,v by A2,ANPROJ_1:1; hence thesis by A1,A3,ANPROJ_1:22; end; uniqueness proof let u,v be non zero Element of TOP-REAL 3 such that A4: P = Dir u & u.3 = 1 and A5: P = Dir v & v.3 = 1; are_Prop u,v by A4,A5,ANPROJ_1:22; then consider a be Real such that a <> 0 and A6: u = a * v by ANPROJ_1:1; A7: 1 = a * v.3 by A4,A6,RVSUM_1:44 .= a by A5; a * v = |[a * v`1, a * v`2, a * v`3]| by EUCLID_5:7 .= v by A7,EUCLID_5:3; hence thesis by A6; end; end; theorem Th17: for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]| proof let P be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; let u9 be non zero Element of TOP-REAL 3; assume P = Dir u9; then Dir u9 = Dir normalize_proj3 P by Def6; then are_Prop u9,normalize_proj3 P by ANPROJ_1:22; then consider a be Real such that a <> 0 and A1: normalize_proj3 P = a * u9 by ANPROJ_1:1; A2: normalize_proj3 P = |[a * u9`1,a * u9`2,a * u9`3 ]| by A1,EUCLID_5:7; A3: 1 = (normalize_proj3 P)`3 by Def6 .= a * u9`3 by A2,EUCLID_5:2; then A4: u9`3 = 1 / a & a = 1 / u9`3 by XCMPLX_1:73; normalize_proj3 P = |[ u9`1 / u9`3,(1 / u9`3) * u9`2,1]| by A1,A3,A4,EUCLID_5:7 .= |[ u9.1 / u9.3,u9.2/u9.3,1]|; hence thesis; end; theorem for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 for Q being Point of ProjectiveSpace TOP-REAL 3 st Q = Dir normalize_proj3(P) holds Q is non zero_proj3 by Def6; registration cluster non zero_proj1 non zero_proj2 for Point of ProjectiveSpace TOP-REAL 3; existence proof reconsider u = |[1,1,0]| as non zero Element of TOP-REAL 3; reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26; take P; reconsider un = u as Element of REAL 3 by EUCLID:22; |[1,1,0]| = |[un.1,un.2,un.3]| by EUCLID_8:1; then u.1 <> 0 & u.2 <> 0 by FINSEQ_1:78; hence thesis; end; end; registration cluster non zero_proj1 non zero_proj3 for Point of ProjectiveSpace TOP-REAL 3; existence proof reconsider u = |[1,0,1]| as non zero Element of TOP-REAL 3; reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26; take P; reconsider un = u as Element of REAL 3 by EUCLID:22; |[1,0,1]| = |[un.1,un.2,un.3]| by EUCLID_8:1; then u.1 <> 0 & u.3 <> 0 by FINSEQ_1:78; hence thesis; end; end; registration cluster non zero_proj2 non zero_proj3 for Point of ProjectiveSpace TOP-REAL 3; existence proof reconsider u = |[0,1,1]| as non zero Element of TOP-REAL 3; reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26; take P; reconsider un = u as Element of REAL 3 by EUCLID:22; |[0,1,1]| = |[un.1,un.2,un.3]| by EUCLID_8:1; then u.2 <> 0 & u.3 <> 0 by FINSEQ_1:78; hence thesis; end; end; registration cluster non zero_proj1 non zero_proj2 non zero_proj3 for Point of ProjectiveSpace TOP-REAL 3; existence proof reconsider u = |[1,1,1]| as non zero Element of TOP-REAL 3; reconsider P = Dir u as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26; take P; reconsider un = u as Element of REAL 3 by EUCLID:22; |[1,1,1]| = |[un.1,un.2,un.3]| by EUCLID_8:1; then u.1 <> 0 & u.2 <> 0 & u.3 <> 0 by FINSEQ_1:78; hence thesis; end; end; definition let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; func dir1a(P) -> non zero Element of TOP-REAL 3 equals |[- (normalize_proj1(P)).2,1,0]|; coherence; end; definition let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; func Pdir1a P -> Point of ProjectiveSpace TOP-REAL 3 equals Dir (dir1a P); coherence by ANPROJ_1:26; end; definition let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; func dir1b(P) -> non zero Element of TOP-REAL 3 equals |[- (normalize_proj1(P)).3,0,1]|; coherence; end; definition let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; func Pdir1b P -> Point of ProjectiveSpace TOP-REAL 3 equals Dir (dir1b P); coherence by ANPROJ_1:26; end; theorem for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 holds dir1a(P) <> dir1b(P) by FINSEQ_1:78; theorem Th20: for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 holds Dir dir1a(P) <> Dir dir1b(P) proof let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; assume Dir dir1a(P) = Dir dir1b(P); then are_Prop dir1a(P),dir1b(P) by ANPROJ_1:22; then consider a be Real such that A1: a <> 0 and A2: dir1a(P) = a * dir1b(P) by ANPROJ_1:1; 0 = (dir1a(P))`3 by EUCLID_5:2 .= a * (dir1b(P))`3 by A2,RVSUM_1:44 .= a * 1 by EUCLID_5:2; hence contradiction by A1; end; theorem Th21: for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 for v being Element of TOP-REAL 3 st u = normalize_proj1 P holds |{ dir1a P,dir1b P,v }| = |(u,v)| proof let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; let v be Element of TOP-REAL 3; assume u = normalize_proj1 P; then A1: u.1 = 1 & P = Dir u by Def2; then normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| by Th11; then (normalize_proj1(P))`2 = u.2/u.1 & (normalize_proj1(P))`3 = u.3/u.1 by EUCLID_5:2; then |{ dir1a P,dir1b P,v }| = |{ |[ -u.2/u.1, 1 , 0 ]|, |[ -u.3/u.1, 0 , 1 ]|, |[ v`1 , v`2, v`3 ]| }| by EUCLID_5:3 .= v`1 - (-u.2/u.1) * v`2 - v`3 * (-u.3/u.1) by Th2 .= (1/u.1) * (u`1 * v`1 + u`2 * v`2 + v`3 * u`3) by A1 .= (1/u.1) * |(u,v)| by EUCLID_5:29; hence thesis by A1; end; theorem for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st u = normalize_proj1 P holds |{ dir1a P,dir1b P,normalize_proj1 P }| = 1 + u.2 * u.2 + u.3 * u.3 proof let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; assume A1: u = normalize_proj1 P; then A2: u.1 = 1 by Def2; reconsider un = u as Element of REAL 3 by EUCLID:22; thus |{ dir1a P,dir1b P,normalize_proj1 P }| = |(un,un)| by A1,Th21 .= u.1 * u.1 + u.2 * u.2 + u.3 * u.3 by EUCLID_8:63 .= 1 + u.2 * u.2 + u.3 * u.3 by A2; end; definition let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; func dir2a(P) -> non zero Element of TOP-REAL 3 equals |[1, - (normalize_proj2(P)).1,0]|; coherence; end; definition let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; func Pdir2a P -> Point of ProjectiveSpace TOP-REAL 3 equals Dir (dir2a P); coherence by ANPROJ_1:26; end; definition let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; func dir2b(P) -> non zero Element of TOP-REAL 3 equals |[0, - (normalize_proj2(P)).3,1]|; coherence; end; definition let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; func Pdir2b P -> Point of ProjectiveSpace TOP-REAL 3 equals Dir (dir2b P); coherence by ANPROJ_1:26; end; theorem for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 holds dir2a(P) <> dir2b(P) by FINSEQ_1:78; theorem Th24: for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 holds Dir dir2a(P) <> Dir dir2b(P) proof let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; assume Dir dir2a(P) = Dir dir2b(P); then are_Prop dir2a(P),dir2b(P) by ANPROJ_1:22; then consider a be Real such that A1: a <> 0 and A2: dir2a(P) = a * dir2b(P) by ANPROJ_1:1; 0 = (dir2a(P))`3 by EUCLID_5:2 .= a * (dir2b(P))`3 by A2,RVSUM_1:44 .= a * 1 by EUCLID_5:2; hence contradiction by A1; end; theorem Th25: for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 for v being Element of TOP-REAL 3 st u = normalize_proj2 P holds |{ dir2a P,dir2b P,v }| = - |(u,v)| proof let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; let v be Element of TOP-REAL 3; assume u = normalize_proj2 P; then A1: u.2 = 1 & P = Dir u by Def4; then normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by Th14; then (normalize_proj2(P))`1 = u.1/u.2 & (normalize_proj2(P))`3 = u.3/u.2 by EUCLID_5:2; then |{ dir2a P,dir2b P,v }| = |{ |[ 1, -u.1/u.2, 0 ]|, |[ 0, -u.3/u.2, 1 ]|, |[ v`1, v`2, v`3 ]| }| by EUCLID_5:3 .= (-u.3/u.2) * v`3 + (-u.1/u.2) * v`1 - v`2 by Th3 .= -(1/u.2) * (u`1 * v`1 + u`2 * v`2 + u`3 * v`3) by A1 .= -(1/u.2) * |(u,v)| by EUCLID_5:29; hence thesis by A1; end; theorem for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st u = normalize_proj2 P holds |{ dir2a P,dir2b P,normalize_proj2 P }| = - (u.1 * u.1 + 1 + u.3 * u.3) proof let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; assume A1: u = normalize_proj2 P; then A2: u.2 = 1 by Def4; reconsider un = u as Element of REAL 3 by EUCLID:22; thus |{ dir2a P,dir2b P,normalize_proj2 P }| = - |(un,un)| by A1,Th25 .= - (u.1 * u.1 + u.2 * u.2 + u.3 * u.3) by EUCLID_8:63 .= - (u.1 * u.1 + 1 + u.3 * u.3) by A2; end; definition let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; func dir3a(P) -> non zero Element of TOP-REAL 3 equals |[1,0,- (normalize_proj3(P)).1]|; coherence; end; definition let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; func Pdir3a P -> Point of ProjectiveSpace TOP-REAL 3 equals Dir (dir3a P); coherence by ANPROJ_1:26; end; definition let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; func dir3b(P) -> non zero Element of TOP-REAL 3 equals |[0,1,- (normalize_proj3(P)).2]|; coherence; end; definition let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; func Pdir3b P -> Point of ProjectiveSpace TOP-REAL 3 equals Dir (dir3b P); coherence by ANPROJ_1:26; end; theorem for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 holds dir3a(P) <> dir3b(P) by FINSEQ_1:78; theorem Th28: for P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 holds Dir dir3a(P) <> Dir dir3b(P) proof let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; assume Dir dir3a(P) = Dir dir3b(P); then are_Prop dir3a(P),dir3b(P) by ANPROJ_1:22; then consider a be Real such that A1: a <> 0 and A2: dir3a(P) = a * dir3b(P) by ANPROJ_1:1; 0 = (dir3a(P))`2 by EUCLID_5:2 .= a * (dir3b(P))`2 by A2,RVSUM_1:44 .= a * 1 by EUCLID_5:2; hence contradiction by A1; end; theorem Th29: for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 for v being Element of TOP-REAL 3 st u = normalize_proj3 P holds |{ dir3a P,dir3b P,v }| = |(u,v)| proof let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; let v be Element of TOP-REAL 3; assume u = normalize_proj3 P; then A1: u.3 = 1 & P = Dir u by Def6; then normalize_proj3 P = |[u.1/u.3, u.2/u.3, 1]| by Th17; then (normalize_proj3(P))`1 = u.1/u.3 & (normalize_proj3(P))`2 = u.2/u.3 by EUCLID_5:2; then |{ dir3a P,dir3b P,v }| = |{ |[ 1, 0, -u.1/u.3 ]|, |[ 0, 1, -u.2/u.3 ]|, |[ v`1, v`2, v`3 ]| }| by EUCLID_5:3 .= v`3 - v`1 * (-u.1/u.3) - v`2 * (-u.2/u.3) by Th4 .= u`1 * v`1 + u`2 * v`2 + u`3 * v`3 by A1 .= |(u,v)| by EUCLID_5:29; hence thesis; end; theorem for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st u = normalize_proj3 P holds |{ dir3a P,dir3b P,normalize_proj3 P }| = u.1 * u.1 + u.2 * u.2 + 1 proof let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; assume A1: u = normalize_proj3 P; then A2: u.3 = 1 by Def6; reconsider un = u as Element of REAL 3 by EUCLID:22; thus |{ dir3a P,dir3b P,normalize_proj3 P }| = |(un,un)| by A1,Th29 .= u.1 * u.1 + u.2 * u.2 + u.3 * u.3 by EUCLID_8:63 .= u.1 * u.1 + u.2 * u.2 + 1 by A2; end; definition let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; func dual1 P -> Element of ProjectiveLines real_projective_plane equals Line(Pdir1a P,Pdir1b P); correctness proof reconsider P1 = Pdir1a P, P2 = Pdir1b P as Point of real_projective_plane; reconsider L = Line(P1,P2) as LINE of real_projective_plane by Th20,COLLSP:def 7; L in {B where B is Subset of real_projective_plane: B is LINE of real_projective_plane}; hence thesis; end; end; definition let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; func dual2 P -> Element of ProjectiveLines real_projective_plane equals Line(Pdir2a P,Pdir2b P); correctness proof reconsider P1 = Pdir2a P, P2 = Pdir2b P as Point of real_projective_plane; reconsider L = Line(P1,P2) as LINE of real_projective_plane by Th24,COLLSP:def 7; L in {B where B is Subset of real_projective_plane: B is LINE of real_projective_plane}; hence thesis; end; end; definition let P being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; func dual3 P -> Element of ProjectiveLines real_projective_plane equals Line(Pdir3a P,Pdir3b P); correctness proof reconsider P1 = Pdir3a P, P2 = Pdir3b P as Point of real_projective_plane; reconsider L = Line(P1,P2) as LINE of real_projective_plane by Th28,COLLSP:def 7; L in {B where B is Subset of real_projective_plane: B is LINE of real_projective_plane}; hence thesis; end; end; theorem for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds normalize_proj1 P = |[1, u.2/u.1, u.3/u.1]| & normalize_proj2 P = |[u.1/u.2, 1, u.3/u.2]| by Th11,Th14; theorem for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 for u being non zero Element of TOP-REAL 3 st P = Dir u holds normalize_proj1 P = u.2/u.1 * normalize_proj2 P & normalize_proj2 P = u.1/u.2 * normalize_proj1 P proof let P be non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; let u be non zero Element of TOP-REAL 3; assume A1: P = Dir u; set r = u.1 / u.2; A2: u.1 <> 0 & u.2 <> 0 by A1,Th10,Th13; A3: (u.1/u.2) * (u.2/u.1) = r * (1 / r) by XCMPLX_1:57 .= 1 by A2,XCMPLX_1:106; Dir normalize_proj1 P = P & Dir normalize_proj2 P = P by Def2,Def4; then are_Prop normalize_proj1 P,normalize_proj2 P by ANPROJ_1:22; then consider a be Real such that a <> 0 and A4: normalize_proj1 P = a * normalize_proj2 P by ANPROJ_1:1; normalize_proj1 P = |[1, u.2/u.1, u.3/u.1]| & normalize_proj2 P = |[u.1/u.2, 1, u.3/u.2]| by A1,Th11,Th14; then A5: |[1, u.2/u.1, u.3/u.1]| = |[ a * (u.1 / u.2),a*1,a * (u.3/u.2)]| by A4,EUCLID_5:8; hence normalize_proj1 P = (u.2/u.1) * normalize_proj2 P by A4,FINSEQ_1:78; (u.1/u.2) * normalize_proj1 P = (u.1/u.2) * ((u.2/u.1) * normalize_proj2 P) by A4,A5,FINSEQ_1:78 .= ((u.1/u.2) * (u.2/u.1)) * normalize_proj2 P by RVSUM_1:49 .= normalize_proj2 P by A3,RVSUM_1:52; hence normalize_proj2 P = (u.1/u.2) * normalize_proj1 P; end; theorem Th33: for P being non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 holds dual1 P = dual2 P proof let P be non zero_proj1 non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; consider u be Element of TOP-REAL 3 such that A1: u is not zero and A2: P = Dir u by ANPROJ_1:26; reconsider u as non zero Element of TOP-REAL 3 by A1; A3: normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| & normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by A2,Th11,Th14; now now let x be object; assume x in Line(Pdir1a P,Pdir1b P); then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that A4: x = P9 and A5: Pdir1a P,Pdir1b P,P9 are_collinear; consider u9 be Element of TOP-REAL 3 such that A6: u9 is non zero and A7: P9 = Dir u9 by ANPROJ_1:26; set a2 = - (normalize_proj1(P)).2, a3 = - (normalize_proj1(P)).3, b1 = u9`1, b2 = u9`2, b3 = u9`3; A8: a2 = - (normalize_proj1(P))`2 .= - u.2/u.1 by A3,EUCLID_5:2; A9: a3 = - (normalize_proj1(P))`3 .= - u.3/u.1 by A3,EUCLID_5:2; 0 = |{ dir1a P,dir1b P,u9 }| by A5,A6,A7,BKMODEL1:1 .= |{ |[a2, 1 , 0]| , |[a3, 0 , 1]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= b1 - a2 * b2 - a3 * b3 by Th2 .= b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A8,A9; then A10: 0 = u.1 * (b1 + u.2/u.1 * b2 + u.3/u.1 * b3) .= u.1 * b1 + u.1 * (u.2 / u.1) * b2 + u.1 * (u.3/u.1) * b3 .= u.1 * b1 + u.2 * b2 + u.1 * (u.3/u.1) * b3 by A2,Th10,XCMPLX_1:87 .= u.1 * b1 + u.2 * b2 + u.3 * b3 by A2,Th10,XCMPLX_1:87; set c2 = - (normalize_proj2(P)).1, c3 = - (normalize_proj2(P)).3; A11: c2 = - (normalize_proj2(P))`1 .= - u.1/u.2 by A3,EUCLID_5:2; A12: c3 = - (normalize_proj2(P))`3 .= - u.3/u.2 by A3,EUCLID_5:2; |{ |[1, c2, 0]|, |[0, c3, 1]|, |[u9`1,u9`2,u9`3]| }| = (- u.1/u.2) * b1 + (-u.3/u.2) * b3 - b2 by A11,A12,Th3; then |{dir2a P,dir2b P,u9}| = (- u.1/u.2) * b1 + (-u.3/u.2) * b3 + (-1) * b2 by EUCLID_5:3 .= (- u.1/u.2) * b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2 by XCMPLX_1:60,A2,Th13 .= (u.1/(-u.2)) * b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2 by XCMPLX_1:188 .= (u.1/(-u.2)) * b1 + (u.3/(-u.2)) * b3 + (-u.2/u.2) * b2 by XCMPLX_1:188 .= (u.1/(-u.2)) * b1 + (u.3/(-u.2)) * b3 + (u.2/(-u.2)) * b2 by XCMPLX_1:188 .= (1 / -u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3) .= 0 by A10; then Pdir2a P,Pdir2b P,P9 are_collinear by A6,A7,BKMODEL1:1; hence x in Line(Pdir2a P,Pdir2b P) by A4; end; hence Line(Pdir1a P,Pdir1b P) c= Line(Pdir2a P,Pdir2b P); now let x be object; assume x in Line(Pdir2a P,Pdir2b P); then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that A13: x = P9 and A14: Pdir2a P,Pdir2b P,P9 are_collinear; consider u9 be Element of TOP-REAL 3 such that A15: u9 is non zero and A16: P9 = Dir u9 by ANPROJ_1:26; set a2 = - (normalize_proj1(P)).2, a3 = - (normalize_proj1(P)).3, b1 = u9`1, b2 = u9`2, b3 = u9`3; set c2 = - (normalize_proj2(P)).1, c3 = - (normalize_proj2(P)).3; A17: a2 = - (normalize_proj1(P))`2 .= - u.2/u.1 by A3,EUCLID_5:2; A18: a3 = - (normalize_proj1(P))`3 .= - u.3/u.1 by A3,EUCLID_5:2; A19: c2 = - (normalize_proj2(P))`1 .= - u.1/u.2 by A3,EUCLID_5:2; A20: c3 = - (normalize_proj2(P))`3 .= - u.3/u.2 by A3,EUCLID_5:2; A21: - u.2 <> 0 by A2,Th13; A22: 0 = |{ dir2a P,dir2b P,u9 }| by A14,A15,A16,BKMODEL1:1 .= |{ |[1, c2 , 0]| , |[0, c3 , 1]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= c3 * b3 + c2 * b1 - b2 by Th3 .= (- u.1/u.2) * b1 + (-u.3/u.2) * b3 + (-1) * b2 by A19,A20 .= (- u.1/u.2) * b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2 by XCMPLX_1:60,A2,Th13 .= (u.1/(-u.2)) * b1 + (-u.3/u.2) * b3 + (-u.2/u.2) * b2 by XCMPLX_1:188 .= (u.1/(-u.2)) * b1 + (u.3/(-u.2)) * b3 + (-u.2/u.2) * b2 by XCMPLX_1:188 .= (u.1/(-u.2)) * b1 + (u.3/(-u.2)) * b3 + (u.2/(-u.2)) * b2 by XCMPLX_1:188 .= (1 / -u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3); A23: u.1/u.1 = 1 by XCMPLX_1:60,A2,Th10; |{dir1a P,dir1b P,u9}| = |{ |[a2, 1 , 0]| , |[a3, 0 , 1]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= b1 - a2 * b2 - a3 * b3 by Th2 .= (u.1/u.1) * b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A17,A18,A23 .= (1/u.1) * (u.1 * b1 + u.2 * b2 + u.3 * b3) .= (1 / u.1) * 0 by A22,A21,XCMPLX_1:6 .= 0; then Pdir1a P,Pdir1b P,P9 are_collinear by A15,A16,BKMODEL1:1; hence x in Line(Pdir1a P,Pdir1b P) by A13; end; hence Line(Pdir2a P,Pdir2b P) c= Line(Pdir1a P,Pdir1b P); end; hence thesis; end; theorem Th34: for P being non zero_proj2 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 holds dual2 P = dual3 P proof let P be non zero_proj2 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; consider u be Element of TOP-REAL 3 such that A1: u is not zero and A2: P = Dir u by ANPROJ_1:26; reconsider u as non zero Element of TOP-REAL 3 by A1; A3: normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]| & normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by A2,Th17,Th14; now now let x be object; assume x in Line(Pdir2a P,Pdir2b P); then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that A4: x = P9 and A5: Pdir2a P,Pdir2b P,P9 are_collinear; consider u9 be Element of TOP-REAL 3 such that A6: u9 is non zero and A7: P9 = Dir u9 by ANPROJ_1:26; set a2 = - (normalize_proj2(P)).1, a3 = - (normalize_proj2(P)).3, b1 = u9`1, b2 = u9`2, b3 = u9`3; A8: a2 = - (normalize_proj2(P))`1 .= - u.1/u.2 by A3,EUCLID_5:2; A9: a3 = - (normalize_proj2(P))`3 .= - u.3/u.2 by A3,EUCLID_5:2; 0 = |{ dir2a P,dir2b P,u9 }| by A5,A6,A7,BKMODEL1:1 .= |{ |[1, a2, 0]| , |[0, a3, 1]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= a2 * b1 + a3 * b3 - b2 by Th3 .= -(u.1/u.2 * b1 + b2 + u.3/u.2 * b3) by A8,A9; then A10: 0 = u.2 * (u.1/u.2 *b1 + b2 + u.3/u.2 * b3) .= u.2 * b2 + u.2 * (u.1 / u.2) * b1 + u.2 * (u.3/u.2) * b3 .= u.2 * b2 + u.1 * b1 + u.2 * (u.3/u.2) * b3 by A2,Th13,XCMPLX_1:87 .= u.2 * b2 + u.1 * b1 + u.3 * b3 by A2,Th13,XCMPLX_1:87; set c2 = - (normalize_proj3(P)).1, c3 = - (normalize_proj3(P)).2; A11: c2 = - (normalize_proj3(P))`1 .= - u.1/u.3 by A3,EUCLID_5:2; A12: c3 = - (normalize_proj3(P))`2 .= - u.2/u.3 by A3,EUCLID_5:2; A13: u.3 / u.3 = 1 by A2,Th16,XCMPLX_1:60; |{ |[1, 0,c2]|, |[0, 1,c3]|, |[u9`1,u9`2,u9`3]| }| = b3 - b1 * (-u.1/u.3) - b2 * (-u.2/u.3) by A11,A12,Th4 .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + b3; then |{dir3a P,dir3b P,u9}| = (u.1 * (1/u.3)) * b1 + (u.2/(u.3)) * b2 + (u.3/u.3) * b3 by A13,EUCLID_5:3 .= (1 /u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3) .= 0 by A10; then Pdir3a P,Pdir3b P,P9 are_collinear by A6,A7,BKMODEL1:1; hence x in Line(Pdir3a P,Pdir3b P) by A4; end; hence Line(Pdir2a P,Pdir2b P) c= Line(Pdir3a P,Pdir3b P); now let x be object; assume x in Line(Pdir3a P,Pdir3b P); then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that A14: x = P9 and A15: Pdir3a P,Pdir3b P,P9 are_collinear; consider u9 be Element of TOP-REAL 3 such that A16: u9 is non zero and A17: P9 = Dir u9 by ANPROJ_1:26; set a2 = - (normalize_proj3(P)).1, a3 = - (normalize_proj3(P)).2, b1 = u9`1, b2 = u9`2, b3 = u9`3; set c2 = - (normalize_proj2(P)).1, c3 = - (normalize_proj2(P)).3; A18: a2 = - (normalize_proj3(P))`1 .= - u.1/u.3 by A3,EUCLID_5:2; A19: a3 = - (normalize_proj3(P))`2 .= - u.2/u.3 by A3,EUCLID_5:2; A20: c2 = - (normalize_proj2(P))`1 .= - u.1/u.2 by A3,EUCLID_5:2; A21: c3 = - (normalize_proj2(P))`3 .= - u.3/u.2 by A3,EUCLID_5:2; A22: 0 = |{ dir3a P,dir3b P,u9 }| by A15,A16,A17,BKMODEL1:1 .= |{ |[1, 0, a2]| , |[0, 1, a3]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= b3 - a2 * b1 - a3 * b2 by Th4 .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + 1 * b3 by A18,A19 .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3 by XCMPLX_1:60,A2,Th16 .= (1 /u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3); A23: u.3 <> 0 by A2,Th16; |{dir2a P,dir2b P,u9}| = |{ |[1 ,c2, 0]| , |[0 ,c3, 1]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= c3 * b3 + c2 * b1 - b2 by Th3 .= (-u.1/u.2) * b1 + (-1) * b2 + (-u.3/u.2) * b3 by A20,A21 .= (-u.1/u.2) * b1 + (-u.2/u.2) * b2 + (-u.3/u.2) * b3 by XCMPLX_1:60,A2,Th13 .= (u.1/-u.2) * b1 + (-u.2/u.2) * b2 + (-u.3/u.2) * b3 by XCMPLX_1:188 .= (u.1/-u.2) * b1 + (u.2/-u.2) * b2 + (-u.3/u.2) * b3 by XCMPLX_1:188 .= (u.1/-u.2) * b1 + (u.2/-u.2) * b2 + (u.3/-u.2) * b3 by XCMPLX_1:188 .= (1/-u.2) * (u.1 * b1 + u.2 * b2 + u.3 * b3) .= (1/-u.2) * 0 by A23,XCMPLX_1:6,A22 .= 0; then Pdir2a P,Pdir2b P,P9 are_collinear by A16,A17,BKMODEL1:1; hence x in Line(Pdir2a P,Pdir2b P) by A14; end; hence Line(Pdir3a P,Pdir3b P) c= Line(Pdir2a P,Pdir2b P); end; hence thesis; end; theorem Th35: for P being non zero_proj1 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 holds dual1 P = dual3 P proof let P be non zero_proj1 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; consider u be Element of TOP-REAL 3 such that A1: u is not zero and A2: P = Dir u by ANPROJ_1:26; reconsider u as non zero Element of TOP-REAL 3 by A1; A3: normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| & normalize_proj3 P = |[u.1/u.3,u.2/u.3,1]| by A2,Th11,Th17; now now let x be object; assume x in Line(Pdir1a P,Pdir1b P); then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that A4: x = P9 and A5: Pdir1a P,Pdir1b P,P9 are_collinear; consider u9 be Element of TOP-REAL 3 such that A6: u9 is non zero and A7: P9 = Dir u9 by ANPROJ_1:26; set a2 = - (normalize_proj1(P)).2, a3 = - (normalize_proj1(P)).3, b1 = u9`1, b2 = u9`2, b3 = u9`3; A8: a2 = - (normalize_proj1(P))`2 .= - u.2/u.1 by A3,EUCLID_5:2; A9: a3 = - (normalize_proj1(P))`3 .= - u.3/u.1 by A3,EUCLID_5:2; 0 = |{ dir1a P,dir1b P,u9 }| by A5,A6,A7,BKMODEL1:1 .= |{ |[a2, 1 , 0]| , |[a3, 0 , 1]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= b1 - a2 * b2 - a3 * b3 by Th2 .= b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A8,A9; then A10: 0 = u.1 * (b1 + u.2/u.1 * b2 + u.3/u.1 * b3) .= u.1 * b1 + u.1 * (u.2 / u.1) * b2 + u.1 * (u.3/u.1) * b3 .= u.1 * b1 + u.2 * b2 + u.1 * (u.3/u.1) * b3 by A2,Th10,XCMPLX_1:87 .= u.1 * b1 + u.2 * b2 + u.3 * b3 by A2,Th10,XCMPLX_1:87; set c2 = - (normalize_proj3(P)).1, c3 = - (normalize_proj3(P)).2; A11: c2 = - (normalize_proj3(P))`1 .= - u.1/u.3 by A3,EUCLID_5:2; A12: c3 = - (normalize_proj3(P))`2 .= - u.2/u.3 by A3,EUCLID_5:2; |{ |[1, 0, c2 ]|, |[0, 1, c3]|, |[u9`1,u9`2,u9`3]| }| = b3 - c2 * b1 - c3 * b2 by Th4; then |{dir3a P,dir3b P,u9}| = (u.1/u.3) * b1 + (u.2/u.3) * b2 + 1 * b3 by A11,A12,EUCLID_5:3 .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3 by XCMPLX_1:60,A2,Th16 .= (1 / u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3) .= 0 by A10; then Pdir3a P,Pdir3b P,P9 are_collinear by A6,A7,BKMODEL1:1; hence x in Line(Pdir3a P,Pdir3b P) by A4; end; hence Line(Pdir1a P,Pdir1b P) c= Line(Pdir3a P,Pdir3b P); now let x be object; assume x in Line(Pdir3a P,Pdir3b P); then consider P9 be Point of ProjectiveSpace TOP-REAL 3 such that A13: x = P9 and A14: Pdir3a P,Pdir3b P,P9 are_collinear; consider u9 be Element of TOP-REAL 3 such that A15: u9 is non zero and A16: P9 = Dir u9 by ANPROJ_1:26; set a2 = - (normalize_proj1(P)).2, a3 = - (normalize_proj1(P)).3, b1 = u9`1, b2 = u9`2, b3 = u9`3; set c2 = - (normalize_proj3(P)).1, c3 = - (normalize_proj3(P)).2; A17: a2 = - (normalize_proj1(P))`2 .= - u.2/u.1 by A3,EUCLID_5:2; A18: a3 = - (normalize_proj1(P))`3 .= - u.3/u.1 by A3,EUCLID_5:2; A19: c2 = - (normalize_proj3(P))`1 .= - u.1/u.3 by A3,EUCLID_5:2; A20: c3 = - (normalize_proj3(P))`2 .= - u.2/u.3 by A3,EUCLID_5:2; A21: 0 = |{ dir3a P,dir3b P,u9 }| by A14,A15,A16,BKMODEL1:1 .= |{ |[1, 0,c2]| , |[0, 1,c3]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= b3 - c2 * b1 - c3 * b2 by Th4 .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + 1 * b3 by A19,A20 .= (u.1/u.3) * b1 + (u.2/u.3) * b2 + (u.3/u.3) * b3 by XCMPLX_1:60,A2,Th16 .= (1 / u.3) * (u.1 * b1 + u.2 * b2 + u.3 * b3); A22: u.3 <> 0 by A2,Th16; A23: u.1/u.1 = 1 by XCMPLX_1:60,A2,Th10; |{dir1a P,dir1b P,u9}| = |{ |[a2, 1 , 0]| , |[a3, 0 , 1]|, |[b1, b2, b3]| }| by EUCLID_5:3 .= b1 - a2 * b2 - a3 * b3 by Th2 .= (u.1/u.1) * b1 + u.2/u.1 * b2 + u.3/u.1 * b3 by A17,A18,A23 .= (1/u.1) * (u.1 * b1 + u.2 * b2 + u.3 * b3) .= (1 / u.1) * 0 by A22,A21,XCMPLX_1:6 .= 0; then Pdir1a P,Pdir1b P,P9 are_collinear by A15,A16,BKMODEL1:1; hence x in Line(Pdir1a P,Pdir1b P) by A13; end; hence Line(Pdir3a P,Pdir3b P) c= Line(Pdir1a P,Pdir1b P); end; hence thesis; end; theorem for P being non zero_proj1 non zero_proj2 non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 holds dual1 P = dual2 P & dual1 P = dual3 P & dual2 P = dual3 P by Th33,Th34,Th35; theorem Th37: for P being Element of ProjectiveSpace TOP-REAL 3 holds P is non zero_proj1 or P is non zero_proj2 or P is non zero_proj3 proof let P be Element of ProjectiveSpace TOP-REAL 3; assume that A1: P is zero_proj1 and A2: P is zero_proj2 and A3: P is zero_proj3; consider u be Element of TOP-REAL 3 such that A4: u is not zero and A5: Dir u = P by ANPROJ_1:26; reconsider u as non zero Element of TOP-REAL 3 by A4; u`1 = 0 & u`2 = 0 & u`3 = 0 by A1,A2,A3,A5; hence thesis by EUCLID_5:3,4; end; definition let P being Point of ProjectiveSpace TOP-REAL 3; func dual P -> Element of ProjectiveLines real_projective_plane means :Def22: ex P9 being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 st P9 = P & it = dual1 P9 if P is non zero_proj1, ex P9 being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 st P9 = P & it = dual2 P9 if (P is zero_proj1 & P is non zero_proj2), ex P9 being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 st P9 = P & it = dual3 P9 if (P is zero_proj1 & P is zero_proj2 & P is non zero_proj3); correctness proof per cases by Th37; suppose P is non zero_proj1; then reconsider P9 = P as non zero_proj1 Element of ProjectiveSpace TOP-REAL 3; dual1 P9 is Element of ProjectiveLines real_projective_plane; hence thesis; end; suppose A1: P is zero_proj1 & P is non zero_proj2; then reconsider P9 = P as non zero_proj2 Element of ProjectiveSpace TOP-REAL 3; dual2 P9 is Element of ProjectiveLines real_projective_plane; hence thesis by A1; end; suppose A3: P is zero_proj1 & P is zero_proj2 & P is non zero_proj3; then reconsider P9 = P as non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; dual3 P9 is Element of ProjectiveLines real_projective_plane; hence thesis by A3; end; end; end; definition let P being Point of real_projective_plane; func # P -> Element of ProjectiveSpace TOP-REAL 3 equals P; coherence; end; definition let P being Point of real_projective_plane; func dual P -> Element of ProjectiveLines real_projective_plane equals dual #P; coherence; end; theorem Th38: for P being Element of real_projective_plane st #P is non zero_proj1 holds ex P9 being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 st P = P9 & dual P = dual1 P9 proof let P be Element of real_projective_plane; assume A1: #P is non zero_proj1; reconsider P1 = #P as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 by A1; per cases; suppose P1 is non zero_proj2 & P1 is zero_proj3; hence thesis by Def22; end; suppose P1 is zero_proj2 & P1 is non zero_proj3; hence thesis by Def22; end; suppose P1 is zero_proj2 & P1 is zero_proj3; hence thesis by Def22; end; suppose P1 is non zero_proj2 & P1 is non zero_proj3; hence thesis by Def22; end; end; theorem Th39: for P being Element of real_projective_plane st #P is non zero_proj2 holds ex P9 being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 st P = P9 & dual P = dual2 P9 proof let P be Element of real_projective_plane; assume A1: #P is non zero_proj2; reconsider P1 = #P as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 by A1; per cases; suppose P1 is non zero_proj1 & P1 is zero_proj3; then reconsider P9 = P1 as non zero_proj1 non zero_proj2 Element of ProjectiveSpace TOP-REAL 3; dual P1 = dual1 P9 & dual1 P9 = dual2 P9 by Def22,Th33; hence thesis; end; suppose P1 is zero_proj1 & P1 is non zero_proj3; hence thesis by Def22; end; suppose P1 is zero_proj1 & P1 is zero_proj3; hence thesis by Def22; end; suppose P1 is non zero_proj1 & P1 is non zero_proj3; then reconsider P9 = P as non zero_proj1 non zero_proj2 non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; dual P1 = dual1 P9 & dual1 P9 = dual2 P9 by Def22,Th33; hence thesis; end; end; theorem Th40: for P being Element of real_projective_plane st #P is non zero_proj3 holds ex P9 being non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 st P = P9 & dual P = dual3 P9 proof let P be Element of real_projective_plane; assume A1: #P is non zero_proj3; reconsider P1 = #P as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 by A1; per cases; suppose A2: P1 is non zero_proj2 & P1 is zero_proj1; then reconsider P9 = P as non zero_proj2 non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; dual P1 = dual2 P9 & dual3 P9 = dual2 P9 by A2,Def22,Th34; hence thesis; end; suppose P1 is zero_proj2 & P1 is non zero_proj1; then reconsider P9 = P as non zero_proj1 non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; dual P1 = dual1 P9 & dual1 P9 = dual3 P9 by Def22,Th35; hence thesis; end; suppose P1 is zero_proj2 & P1 is zero_proj1; hence thesis by Def22; end; suppose P1 is non zero_proj2 & P1 is non zero_proj1; then reconsider P9 = P as non zero_proj1 non zero_proj2 non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; dual P1 = dual1 P9 & dual1 P9 = dual3 P9 by Th35,Def22; hence thesis; end; end; theorem Th41: for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3 holds not P in Line(Pdir1a P,Pdir1b P) proof let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3; assume P in Line(Pdir1a P,Pdir1b P); then A1: Pdir1a P,Pdir1b P,P are_collinear by COLLSP:11; reconsider u = normalize_proj1 P as non zero Element of TOP-REAL 3; A2: P = Dir u by Def2; |{ dir1a P,dir1b P,u }| = |( u, u )| by Th21; then |(u, u)| = 0 by A2,A1,BKMODEL1:1; hence thesis by Th5; end; theorem Th42: for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3 holds not P in Line(Pdir2a P,Pdir2b P) proof let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3; assume P in Line(Pdir2a P,Pdir2b P); then A1: Pdir2a P,Pdir2b P, P are_collinear by COLLSP:11; reconsider u = normalize_proj2 P as non zero Element of TOP-REAL 3; A2: P = Dir u by Def4; |{ dir2a P,dir2b P,u }| = - |( u, u )| by Th25; then |(u, u)| = 0 by A2,A1,BKMODEL1:1; hence thesis by Th5; end; theorem Th43: for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3 holds not P in Line(Pdir3a P,Pdir3b P) proof let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; assume P in Line(Pdir3a P,Pdir3b P); then A1: Pdir3a P,Pdir3b P, P are_collinear by COLLSP:11; reconsider u = normalize_proj3 P as non zero Element of TOP-REAL 3; A2: P = Dir u by Def6; |{ dir3a P,dir3b P,u }| = |( u, u )| by Th29; then |(u, u)| = 0 by A2,A1,BKMODEL1:1; hence thesis by Th5; end; theorem for P being Point of real_projective_plane holds not P in dual P proof let P be Point of real_projective_plane; reconsider P9 = P as Element of ProjectiveSpace TOP-REAL 3; per cases by Th37; suppose P9 is non zero_proj1; then reconsider P9 = P as non zero_proj1 Element of ProjectiveSpace TOP-REAL 3; #P = P9; then consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 such that A1: P = P99 and A2: dual P = dual1 P99 by Th38; assume P in dual P; hence contradiction by A1,A2,Th41; end; suppose P9 is non zero_proj2; then reconsider P9 = P as non zero_proj2 Element of ProjectiveSpace TOP-REAL 3; #P = P9; then consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 such that A3: P = P99 and A4: dual P = dual2 P99 by Th39; assume P in dual P; hence contradiction by Th42,A3,A4; end; suppose P9 is non zero_proj3; then reconsider P9 = P as non zero_proj3 Element of ProjectiveSpace TOP-REAL 3; #P = P9; then consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 such that A5: P = P99 and A6: dual P = dual3 P99 by Th40; assume P in dual P; hence contradiction by Th43,A5,A6; end; end; definition let l being Element of ProjectiveLines real_projective_plane; func dual l -> Point of real_projective_plane means :Def25: ex P,Q being Point of real_projective_plane st P <> Q & l = Line(P,Q) & it = L2P(P,Q); existence proof consider P,Q be Point of real_projective_plane such that A1: P <> Q and A2: l = Line(P,Q) by BKMODEL1:72; L2P(P,Q) is Point of real_projective_plane; hence thesis by A1,A2; end; uniqueness proof let P1,P2 be Point of real_projective_plane such that A3: ex P,Q be Point of real_projective_plane st P <> Q & l = Line(P,Q) & P1 = L2P(P,Q) and A4: ex P,Q being Point of real_projective_plane st P <> Q & l = Line(P,Q) & P2 = L2P(P,Q); consider P,Q be Point of real_projective_plane such that A5: P <> Q and A6: l = Line(P,Q) and A7: P1 = L2P(P,Q) by A3; consider u,v be non zero Element of TOP-REAL 3 such that A8: P = Dir u and A9: Q = Dir v and A10: L2P(P,Q) = Dir(u v) by A5,BKMODEL1:def 5; consider P9,Q9 be Point of real_projective_plane such that A11: P9 <> Q9 and A12: l = Line(P9,Q9) and A13: P2 = L2P(P9,Q9) by A4; consider u9,v9 be non zero Element of TOP-REAL 3 such that A14: P9 = Dir u9 and A15: Q9 = Dir v9 and A16: L2P(P9,Q9) = Dir(u9 v9) by A11,BKMODEL1:def 5; P,Q,P9 are_collinear & P,Q,Q9 are_collinear by A6,A12,COLLSP:10,11; then |{u,v,u9}| = 0 & |{u,v,v9}| = 0 by A8,A9,A14,A15,BKMODEL1:1; then A17: |{u9,u,v}| = 0 & |{v9,u,v}| = 0 by EUCLID_5:33; A18: now now assume u v = 0.TOP-REAL 3; then are_Prop u,v by ANPROJ_8:51; hence contradiction by A8,A9,A5,ANPROJ_1:22; end; hence u v is non zero; now assume u9 v9 = 0.TOP-REAL 3; then are_Prop u9,v9 by ANPROJ_8:51; hence contradiction by A14,A15,A11,ANPROJ_1:22; end; hence u9 v9 is non zero; end; then reconsider uv = u v, u9v9 = u9 v9 as non zero Element of TOP-REAL 3; not are_Prop u9,v9 by A11,A14,A15,ANPROJ_1:22; then are_Prop uv,u9 v9 by A17,Th8; hence thesis by A18,ANPROJ_1:22,A7,A13,A10,A16; end; end; theorem Th45: for P being Point of real_projective_plane holds dual dual P = P proof let P be Point of real_projective_plane; reconsider P9 = P as Point of ProjectiveSpace TOP-REAL 3; per cases by Th37; suppose P9 is non zero_proj1; then reconsider P9 = P as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; reconsider P = P9 as Point of real_projective_plane; consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 such that A1: P = P99 & dual P= dual1 P99 by Th38; reconsider l = Line(Pdir1a P9,Pdir1b P9) as Element of ProjectiveLines real_projective_plane by A1; consider P1,P2 be Point of real_projective_plane such that A2: P1 <> P2 and A3: l = Line(P1,P2) and A4: dual l = L2P(P1,P2) by Def25; A5: Line(P1,P2) = Line(Pdir1a P9,Pdir1b P9) & P1 in Line(P1,P2) & P2 in Line(P1,P2) by A3,COLLSP:10; then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that A6: P1 = Q1 and A7: Pdir1a P9,Pdir1b P9,Q1 are_collinear; consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that A8: P2 = Q2 and A9: Pdir1a P9,Pdir1b P9,Q2 are_collinear by A5; consider u,v be non zero Element of TOP-REAL 3 such that A10: P1 = Dir u and A11: P2 = Dir v and A12: L2P(P1,P2) = Dir(u v) by A2,BKMODEL1:def 5; consider w be Element of TOP-REAL 3 such that A13: w is not zero and A14: P9 = Dir w by ANPROJ_1:26; reconsider w as non zero Element of TOP-REAL 3 by A13; normalize_proj1 P9 = |[1, w.2/w.1,w.3/w.1]| by A14,Th11; then (normalize_proj1(P9))`2 = w.2/w.1 & (normalize_proj1(P9))`3 = w.3/w.1 by EUCLID_5:2; then A15: dir1a P9 dir1b P9 = |[ (1 * 1) - (0 * 0), (0 * (-w.3/w.1)) - ((-w.2/w.1) * 1), ((-w.2/w.1) * 0) - ((-w.3/w.1) * 1) ]| by EUCLID_5:15 .= |[ w`1/w.1, w`2/w.1,w`3/w.1 ]| by A14,Th10,XCMPLX_1:60 .= 1/w.1 * w by EUCLID_5:7; w.1 <> 0 by A14,Th10; then reconsider a = 1/w.1 * w as non zero Element of TOP-REAL 3 by ANPROJ_9:3; now assume u v = 0.TOP-REAL 3; then are_Prop u,v by ANPROJ_8:51; hence contradiction by A2,A10,A11,ANPROJ_1:22; end; then A16: u v is non zero; now now thus not are_Prop u,v by A2,A10,A11,ANPROJ_1:22; thus 0 = |{ dir1a P9,dir1b P9,u }| by A10,A6,A7,BKMODEL1:1 .= |{ u, dir1a P9,dir1b P9 }| by EUCLID_5:34 .= |( a, u )| by A15; thus 0 = |{ dir1a P9,dir1b P9,v }| by A11,A8,A9,BKMODEL1:1 .= |{ v, dir1a P9,dir1b P9 }| by EUCLID_5:34 .= |( a, v )| by A15; end; then are_Prop 1/w.1 * w, u v by Th8; hence are_Prop w.1 * a,u v by A14,Th10,A16,Th9; thus w.1 * a = (w.1 * (1/w.1)) * w by RVSUM_1:49 .= 1 * w by A14,Th10,XCMPLX_1:106 .= w by RVSUM_1:52; end; hence thesis by A14,A1,A4,A12,A16,ANPROJ_1:22; end; suppose P9 is non zero_proj2; then reconsider P9 = P as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; reconsider P = P9 as Point of real_projective_plane; consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 such that A17: P = P99 & dual P= dual2 P99 by Th39; reconsider l = Line(Pdir2a P9,Pdir2b P9) as Element of ProjectiveLines real_projective_plane by A17; consider P1,P2 be Point of real_projective_plane such that A18: P1 <> P2 and A19: l = Line(P1,P2) and A20: dual l = L2P(P1,P2) by Def25; A21: Line(P1,P2) = Line(Pdir2a P9,Pdir2b P9) & P1 in Line(P1,P2) & P2 in Line(P1,P2) by A19,COLLSP:10; then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that A22: P1 = Q1 and A23: Pdir2a P9,Pdir2b P9,Q1 are_collinear; consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that A24: P2 = Q2 and A25: Pdir2a P9,Pdir2b P9,Q2 are_collinear by A21; consider u,v be non zero Element of TOP-REAL 3 such that A26: P1 = Dir u and A27: P2 = Dir v and A28: L2P(P1,P2) = Dir(u v) by A18,BKMODEL1:def 5; consider w be Element of TOP-REAL 3 such that A29: w is not zero and A30: P9 = Dir w by ANPROJ_1:26; reconsider w as non zero Element of TOP-REAL 3 by A29; normalize_proj2 P9 = |[w.1/w.2, 1, w.3/w.2]| by A30,Th14; then (normalize_proj2(P9))`1 = w.1/w.2 & (normalize_proj2(P9))`3 = w.3/w.2 by EUCLID_5:2; then A31: dir2a P9 dir2b P9 = |[ ((-w.1/w.2) * 1) - (0 * (-w.3/w.2)), (0 * 0) - (1 * 1), (1 * (-w.3/w.2)) - (0 * (-w.1/w.2)) ]| by EUCLID_5:15 .= |[ -w.1/w.2, -w.2/w.2,-w.3/w.2 ]| by A30,Th13,XCMPLX_1:60 .= |[ w.1/(-w.2), -w.2/w.2,-w.3/w.2 ]| by XCMPLX_1:188 .= |[ w.1/(-w.2), w.2/(-w.2),-w.3/w.2 ]| by XCMPLX_1:188 .= |[ w`1/(-w.2), w`2/(-w.2),w`3/(-w.2) ]| by XCMPLX_1:188 .= 1/(-w.2) * w by EUCLID_5:7; A32: w.2 <> 0 by A30,Th13; then reconsider a = 1/(-w.2) * w as non zero Element of TOP-REAL 3 by ANPROJ_9:3; now assume u v = 0.TOP-REAL 3; then are_Prop u,v by ANPROJ_8:51; hence contradiction by A18,A26,A27,ANPROJ_1:22; end; then A33: u v is non zero; now now thus not are_Prop u,v by A18,A26,A27,ANPROJ_1:22; thus 0 = |{ dir2a P9,dir2b P9,u }| by A26,A22,A23,BKMODEL1:1 .= |{ u, dir2a P9,dir2b P9 }| by EUCLID_5:34 .= |( a, u )| by A31; thus 0 = |{ dir2a P9,dir2b P9,v }| by A27,A24,A25,BKMODEL1:1 .= |{ v, dir2a P9,dir2b P9 }| by EUCLID_5:34 .= |( a, v )| by A31; end; then are_Prop 1/(-w.2) * w, u v by Th8; hence are_Prop (-w.2) * a,u v by A32,A33,Th9; thus (-w.2) * a = ((-w.2) * (1/(-w.2))) * w by RVSUM_1:49 .= 1 * w by A32,XCMPLX_1:106 .= w by RVSUM_1:52; end; hence thesis by A30,A17,A20,A28,A33,ANPROJ_1:22; end; suppose P9 is non zero_proj3; then reconsider P9 = P as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; reconsider P = P9 as Point of real_projective_plane; consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 such that A34: P = P99 & dual P= dual3 P99 by Th40; reconsider l = Line(Pdir3a P9,Pdir3b P9) as Element of ProjectiveLines real_projective_plane by A34; consider P1,P2 be Point of real_projective_plane such that A35: P1 <> P2 and A36: l = Line(P1,P2) and A37: dual l = L2P(P1,P2) by Def25; A38: Line(P1,P2) = Line(Pdir3a P9,Pdir3b P9) & P1 in Line(P1,P2) & P2 in Line(P1,P2) by A36,COLLSP:10; then consider Q1 be Point of ProjectiveSpace TOP-REAL 3 such that A39: P1 = Q1 and A40: Pdir3a P9,Pdir3b P9,Q1 are_collinear; consider Q2 be Point of ProjectiveSpace TOP-REAL 3 such that A41: P2 = Q2 and A42: Pdir3a P9,Pdir3b P9,Q2 are_collinear by A38; consider u,v be non zero Element of TOP-REAL 3 such that A43: P1 = Dir u and A44: P2 = Dir v and A45: L2P(P1,P2) = Dir(u v) by A35,BKMODEL1:def 5; consider w be Element of TOP-REAL 3 such that A46: w is not zero and A47: P9 = Dir w by ANPROJ_1:26; reconsider w as non zero Element of TOP-REAL 3 by A46; normalize_proj3 P9 = |[w.1/w.3, w.2/w.3, 1]| by A47,Th17; then (normalize_proj3(P9))`1 = w.1/w.3 & (normalize_proj3(P9))`2 = w.2/w.3 by EUCLID_5:2; then A48: dir3a P9 dir3b P9 = |[ (0 * (-w.2/w.3)) - ((-w.1/w.3) * 1), ((-w.1/w.3) * 0) - (1 * (-w.2/w.3)), (1 * 1) - (0 * 0) ]| by EUCLID_5:15 .= |[ w`1/w.3, w`2/w.3,w`3/w.3 ]| by A47,Th16,XCMPLX_1:60 .= 1/(w.3) * w by EUCLID_5:7; w.3 <> 0 by A47,Th16; then reconsider a = 1/(w.3) * w as non zero Element of TOP-REAL 3 by ANPROJ_9:3; now assume u v = 0.TOP-REAL 3; then are_Prop u,v by ANPROJ_8:51; hence contradiction by A35,A43,A44,ANPROJ_1:22; end; then A49: u v is non zero; now now thus not are_Prop u,v by A35,A43,A44,ANPROJ_1:22; thus 0 = |{ dir3a P9,dir3b P9,u }| by A43,A39,A40,BKMODEL1:1 .= |{ u, dir3a P9,dir3b P9 }| by EUCLID_5:34 .= |( a, u )| by A48; thus 0 = |{ dir3a P9,dir3b P9,v }| by A44,A41,A42,BKMODEL1:1 .= |{ v, dir3a P9,dir3b P9 }| by EUCLID_5:34 .= |( a, v )| by A48; end; then are_Prop 1/(w.3) * w, u v by Th8; hence are_Prop (w.3) * a,u v by A47,Th16,A49,Th9; thus w.3 * a = (w.3 * (1/w.3)) * w by RVSUM_1:49 .= 1 * w by A47,Th16,XCMPLX_1:106 .= w by RVSUM_1:52; end; hence thesis by A47,A34,A37,A45,A49,ANPROJ_1:22; end; end; theorem Th46: for l being Element of ProjectiveLines real_projective_plane holds dual dual l = l proof let l be Element of ProjectiveLines real_projective_plane; consider P,Q be Point of real_projective_plane such that A1: P <> Q and A2: l = Line(P,Q) and A3: dual l = L2P(P,Q) by Def25; reconsider P9 = P,Q9 = Q as Point of ProjectiveSpace TOP-REAL 3; consider u,v be non zero Element of TOP-REAL 3 such that A4: P = Dir u and A5: Q = Dir v and A6: L2P(P,Q) = Dir(u v) by A1,BKMODEL1:def 5; reconsider l2 = Line(P,Q) as LINE of real_projective_plane by A1,COLLSP:def 7; not are_Prop u,v by A1,A4,A5,ANPROJ_1:22; then u v is non zero by ANPROJ_8:51; then reconsider uv = u v as non zero Element of TOP-REAL 3; reconsider R = Dir uv as Point of ProjectiveSpace TOP-REAL 3 by ANPROJ_1:26; reconsider R9 = R as Element of real_projective_plane; A7: 0 = |( u v,u )| by ANPROJ_8:44 .= uv`1 * u`1 + uv`2 * u`2 + uv`3 * u`3 by EUCLID_5:29 .= uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3; A8: 0 = |( u v,v )| by ANPROJ_8:45 .= uv`1 * v`1 + uv`2 * v`2 + uv`3 * v`3 by EUCLID_5:29 .= uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3; per cases by Th37; suppose A9: R is non zero_proj1; then reconsider R as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; #R9 = R; then consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 such that A10: R9 = P99 and A11: dual R9 = dual1 P99 by Th38; A12: uv.1 / uv.1 = 1 by A9,Th10,XCMPLX_1:60; normalize_proj1 P99 = |[1, uv.2/uv.1,uv.3/uv.1]| by A10,Th11; then A13: (normalize_proj1 P99)`2 = uv.2/uv.1 & (normalize_proj1 P99)`3 = uv.3/uv.1 by EUCLID_5:2; reconsider l1 = Line(Pdir1a P99,Pdir1b P99) as LINE of real_projective_plane by Th20,COLLSP:def 7; now |{ |[- uv.2/uv.1, 1, 0]|, |[- uv.3/uv.1, 0, 1]|, |[u`1, u`2,u`3]| }| = u`1 - (-uv.2/uv.1) * u`2 - u`3 * (-uv.3/uv.1) by Th2 .= (uv.1 / uv.1) * u`1 + (uv.2/uv.1) * u`2 + u`3 * (uv.3/uv.1) by A12 .= (1/uv.1) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3) .= 0 by A7; then|{ dir1a P99,dir1b P99,u }| = 0 by A13,EUCLID_5:3; then Pdir1a P99,Pdir1b P99, P9 are_collinear by A4,BKMODEL1:1; hence P in l1; |{ |[- uv.2/uv.1, 1, 0]|, |[- uv.3/uv.1, 0, 1]|, |[v`1, v`2,v`3]| }| = (uv.1 / uv.1) * v`1 - (-uv.2/uv.1) * v`2 - v`3 * (-uv.3/uv.1) by A12,Th2 .= (1/uv.1) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3) .= 0 by A8; then|{ dir1a P99,dir1b P99,v }| = 0 by A13,EUCLID_5:3; then Pdir1a P99,Pdir1b P99, Q9 are_collinear by A5,BKMODEL1:1; hence Q in l1; thus P in l2 & Q in l2 by COLLSP:10; end; hence thesis by A1,A3,A6,A11,A2,COLLSP:20; end; suppose A14: R is non zero_proj2; then reconsider R as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; #R9 = R; then consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 such that A15: R9 = P99 and A16: dual R9 = dual2 P99 by Th39; A17: uv.2 / uv.2 = 1 by A14,Th13,XCMPLX_1:60; normalize_proj2 P99 = |[uv.1/uv.2,1,uv.3/uv.2]| by A15,Th14; then A18: (normalize_proj2 P99)`1 = uv.1/uv.2 & (normalize_proj2 P99)`3 = uv.3/uv.2 by EUCLID_5:2; reconsider l1 = Line(Pdir2a P99,Pdir2b P99) as LINE of real_projective_plane by Th24,COLLSP:def 7; now |{ |[1 , - uv.1/uv.2, 0]|, |[0 , - uv.3/uv.2, 1]|, |[u`1, u`2, u`3]| }| = (-uv.3/uv.2) * u`3 + (-uv.1/uv.2) * u`1 - u`2 by Th3 .= -((uv.3/uv.2) * u`3 + (uv.1/uv.2) * u`1 + (uv.2/uv.2) * u`2) by A17 .= - ((1 / uv.2) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3)) .= 0 by A7; then|{ dir2a P99,dir2b P99,u }| = 0 by A18,EUCLID_5:3; then Pdir2a P99,Pdir2b P99, P9 are_collinear by A4,BKMODEL1:1; hence P in l1; |{ |[ 1 , - uv.1/uv.2, 0 ]|, |[ 0 , - uv.3/uv.2, 1 ]|, |[v`1, v`2, v`3]| }| = (-uv.3/uv.2) * v`3 + (-uv.1/uv.2) * v`1 - v`2 by Th3 .= -((uv.3/uv.2) * v`3 + (uv.1/uv.2) * v`1 + (uv.2/uv.2) * v`2) by A17 .= - ((1 / uv.2) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3)) .= 0 by A8; then|{ dir2a P99,dir2b P99,v }| = 0 by A18,EUCLID_5:3; then Pdir2a P99,Pdir2b P99, Q9 are_collinear by A5,BKMODEL1:1; hence Q in l1; thus P in l2 & Q in l2 by COLLSP:10; end; hence thesis by A3,A6,A16,A2,A1,COLLSP:20; end; suppose A19: R is non zero_proj3; then reconsider R as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; #R9 = R; then consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 such that A20: R9 = P99 and A21: dual R9 = dual3 P99 by Th40; A22: uv.3 / uv.3 = 1 by A19,Th16,XCMPLX_1:60; normalize_proj3 P99 = |[uv.1/uv.3,uv.2/uv.3,1]| by A20,Th17; then A23: (normalize_proj3 P99)`1 = uv.1/uv.3 & (normalize_proj3 P99)`2 = uv.2/uv.3 by EUCLID_5:2; reconsider l1 = Line(Pdir3a P99,Pdir3b P99) as LINE of real_projective_plane by Th28,COLLSP:def 7; now |{ |[1 ,0 , - uv.1/uv.3]|, |[0 ,1 , - uv.2/uv.3]|, |[u`1,u`2, u`3 ]| }| = u`3 - u`1 * (-uv.1/uv.3) - u`2 * (-uv.2/uv.3) by Th4 .= u`1 * (uv.1/uv.3) + u`2 * (uv.2/uv.3) + u`3 * (uv.3 / uv.3) by A22 .= (1 / uv.3) * (uv.1 * u`1 + uv.2 * u`2 + uv.3 * u`3) .= 0 by A7; then|{ dir3a P99,dir3b P99,u }| = 0 by A23,EUCLID_5:3; then Pdir3a P99,Pdir3b P99, P9 are_collinear by A4,BKMODEL1:1; hence P in l1; |{ |[1 ,0 ,- uv.1/uv.3]|, |[0 ,1 ,- uv.2/uv.3]|, |[v`1,v`2,v`3]| }| = v`3 - v`1 * (-uv.1/uv.3) - v`2 * (-uv.2/uv.3) by Th4 .= v`1 * (uv.1/uv.3) + v`2 * (uv.2/uv.3) + v`3 * (uv.3 / uv.3) by A22 .= (1 / uv.3) * (uv.1 * v`1 + uv.2 * v`2 + uv.3 * v`3) .= 0 by A8; then|{ dir3a P99,dir3b P99,v }| = 0 by A23,EUCLID_5:3; then Pdir3a P99,Pdir3b P99, Q9 are_collinear by A5,BKMODEL1:1; hence Q in l1; thus P in l2 & Q in l2 by COLLSP:10; end; hence thesis by A3,A6,A21,A2,A1,COLLSP:20; end; end; theorem for P,Q being Point of real_projective_plane holds (P <> Q iff dual P <> dual Q) proof let P,Q be Point of real_projective_plane; now assume A1: P <> Q; assume dual P = dual Q; then P = dual dual Q by Th45; hence contradiction by A1,Th45; end; hence thesis; end; theorem Th48: for l,m being Element of ProjectiveLines real_projective_plane holds (l <> m iff dual l <> dual m) proof let l,m be Element of ProjectiveLines real_projective_plane; now assume A1: l <> m; assume dual l = dual m; then l = dual dual m by Th46; hence contradiction by A1,Th46; end; hence thesis; end; begin definition let l1,l2,l3 being Element of ProjectiveLines real_projective_plane; pred l1,l2,l3 are_concurrent means ex P being Point of real_projective_plane st P in l1 & P in l2 & P in l3; end; definition let l being Element of ProjectiveLines real_projective_plane; func #l -> LINE of IncProjSp_of real_projective_plane equals l; coherence; end; definition let l being LINE of IncProjSp_of real_projective_plane; func #l -> Element of ProjectiveLines real_projective_plane equals l; coherence; end; theorem Th49: for l1,l2,l3 being Element of ProjectiveLines real_projective_plane holds l1,l2,l3 are_concurrent iff #l1,#l2,#l3 are_concurrent proof let l1,l2,l3 be Element of ProjectiveLines real_projective_plane; now l1 in {B where B is Subset of real_projective_plane: B is LINE of real_projective_plane}; hence ex B be Subset of real_projective_plane st l1 = B & B is LINE of real_projective_plane; l2 in {B where B is Subset of real_projective_plane: B is LINE of real_projective_plane}; hence ex B be Subset of real_projective_plane st l2 = B & B is LINE of real_projective_plane; l3 in {B where B is Subset of real_projective_plane: B is LINE of real_projective_plane}; hence ex B be Subset of real_projective_plane st l3 = B & B is LINE of real_projective_plane; end; then reconsider m1 = l1,m2 = l2,m3 = l3 as LINE of real_projective_plane; reconsider l91 = #l1, l92 = #l2, l93 = #l3 as LINE of IncProjSp_of real_projective_plane; hereby assume l1,l2,l3 are_concurrent; then consider P be Point of real_projective_plane such that A1: P in l1 and A2: P in l2 and A3: P in l3; reconsider P as Element of the Points of IncProjSp_of real_projective_plane; reconsider P9 = P as POINT of IncProjSp_of real_projective_plane; P in m1 & P in m2 & P in m3 by A1,A2,A3; then P9 on l91 & P9 on l92 & P9 on l93 by INCPROJ:5; hence #l1,#l2,#l3 are_concurrent; end; assume #l1,#l2,#l3 are_concurrent; then consider o be Element of the Points of IncProjSp_of real_projective_plane such that A4: o on #l1 and A5: o on #l2 and A6: o on #l3; reconsider o9 = o as Point of real_projective_plane; o9 in m1 & o9 in m2 & o9 in m3 by A4,A5,A6,INCPROJ:5; hence l1,l2,l3 are_concurrent; end; theorem for l1,l2,l3 being LINE of IncProjSp_of real_projective_plane holds l1,l2,l3 are_concurrent iff #l1,#l2,#l3 are_concurrent proof let l1,l2,l3 be LINE of IncProjSp_of real_projective_plane; reconsider l91 = #l1, l92 = #l2, l93 = #l3 as Element of ProjectiveLines real_projective_plane; hereby assume l1,l2,l3 are_concurrent; then #l91,#l92,#l93 are_concurrent; hence #l1,#l2,#l3 are_concurrent by Th49; end; assume #l1,#l2,#l3 are_concurrent; then ##l1,##l2,##l3 are_concurrent by Th49; hence l1,l2,l3 are_concurrent; end; theorem for P,Q,R being Element of real_projective_plane st P,Q,R are_collinear holds Q,R,P are_collinear & R,P,Q are_collinear & P,R,Q are_collinear & R,Q,P are_collinear & Q,P,R are_collinear by ANPROJ_2:24; theorem for l1,l2,l3 being Element of ProjectiveLines real_projective_plane st l1,l2,l3 are_concurrent holds l2,l1,l3 are_concurrent & l1,l3,l2 are_concurrent & l3,l2,l1 are_concurrent & l3,l2,l1 are_concurrent & l2,l3,l1 are_concurrent; theorem for P,Q being Point of real_projective_plane for P9,Q9 being Element of ProjectiveSpace TOP-REAL 3 st P = P9 & Q = Q9 holds Line(P,Q) = Line(P9,Q9); theorem Th54: for P being Point of real_projective_plane for l being Element of ProjectiveLines real_projective_plane st P in l holds dual l in dual P proof let P be Point of real_projective_plane; let l be Element of ProjectiveLines real_projective_plane; assume A1: P in l; consider u be Element of TOP-REAL 3 such that A2: u is not zero and A3: P = Dir u by ANPROJ_1:26; reconsider u as non zero Element of TOP-REAL 3 by A2; reconsider P9 = P as Element of ProjectiveSpace TOP-REAL 3; reconsider dl = dual l as Point of ProjectiveSpace TOP-REAL 3; consider Pl,Ql be Point of real_projective_plane such that A4: Pl <> Ql and A5: l = Line(Pl,Ql) and A6: dual l = L2P(Pl,Ql) by Def25; consider ul,vl be non zero Element of TOP-REAL 3 such that A7: Pl = Dir ul and A8: Ql = Dir vl and A9: L2P(Pl,Ql) = Dir(ul vl) by A4,BKMODEL1:def 5; reconsider ulvl = ul vl as non zero Element of TOP-REAL 3 by A4,A7,A8,BKMODEL1:78; consider S be Point of real_projective_plane such that A10: P = S and A11: Pl,Ql,S are_collinear by A1,A5; P,Pl,Ql are_collinear by A10,A11,ANPROJ_2:24; then A12: |{u,ul,vl}| = 0 by A3,A7,A8,BKMODEL1:1; per cases by Th37; suppose P9 is non zero_proj1; then reconsider P9 as non zero_proj1 Point of ProjectiveSpace TOP-REAL 3; consider P99 be non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 such that A13: P9 = P99 and A14: dual P = dual1 P99 by Th38; consider S be Point of real_projective_plane such that A15: P = S and A16: Pl,Ql,S are_collinear by A1,A5; P,Pl,Ql are_collinear by A15,A16,ANPROJ_2:24; then A17: |{u,ul,vl}| = 0 by A3,A7,A8,BKMODEL1:1; Dir normalize_proj1 P9 = Dir u by A3,Def2; then A18: are_Prop normalize_proj1 P9,u by ANPROJ_1:22; |{dir1a P9,dir1b P9, ulvl }| = |(normalize_proj1 P9, ulvl)| by Th21 .= 0 by A17,A18,Th7; then Pdir1a P9,Pdir1b P9, dl are_collinear by A6,A9,BKMODEL1:1; hence thesis by A13,A14; end; suppose P9 is non zero_proj2; then reconsider P9 as non zero_proj2 Point of ProjectiveSpace TOP-REAL 3; consider P99 be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 such that A19: P9 = P99 and A20: dual P = dual2 P99 by Th39; Dir normalize_proj2 P9 = Dir u by A3,Def4; then A21: are_Prop normalize_proj2 P9,u by ANPROJ_1:22; |{dir2a P9,dir2b P9, ulvl }| = - |(normalize_proj2 P9, ulvl)| by Th25 .= - 0 by A12,A21,Th7; then Pdir2a P9,Pdir2b P9, dl are_collinear by A6,A9,BKMODEL1:1; hence thesis by A19,A20; end; suppose P9 is non zero_proj3; then reconsider P9 as non zero_proj3 Point of ProjectiveSpace TOP-REAL 3; consider P99 be non zero_proj3 Point of ProjectiveSpace TOP-REAL 3 such that A22: P9 = P99 and A23: dual P = dual3 P99 by Th40; Dir normalize_proj3 P9 = Dir u by A3,Def6; then A24: are_Prop normalize_proj3 P9,u by ANPROJ_1:22; |{dir3a P9,dir3b P9, ulvl }| = |(normalize_proj3 P9, ulvl)| by Th29 .= 0 by A12,A24,Th7; then Pdir3a P9,Pdir3b P9, dl are_collinear by A6,A9,BKMODEL1:1; hence thesis by A22,A23; end; end; theorem for P being Point of real_projective_plane for l being Element of ProjectiveLines real_projective_plane st dual l in dual P holds P in l proof let P be Point of real_projective_plane; let l be Element of ProjectiveLines real_projective_plane; assume dual l in dual P; then dual dual P in dual dual l by Th54; then P in dual dual l by Th45; hence thesis by Th46; end; theorem Th56: for P,Q,R being Point of real_projective_plane st P,Q,R are_collinear holds dual P,dual Q, dual R are_concurrent proof let P,Q,R be Point of real_projective_plane; assume A1: P,Q,R are_collinear; per cases; suppose A2: Q = R; reconsider lP = dual P,lQ = dual Q as LINE of real_projective_plane by INCPROJ:1; ex x be object st x in lP & x in lQ by BKMODEL1:76,XBOOLE_0:3; hence thesis by A2; end; suppose A3: Q <> R; A4: Q,R,P are_collinear by A1,ANPROJ_2:24; reconsider l = Line(Q,R) as LINE of real_projective_plane by A3,COLLSP:def 7; l in {B where B is Subset of real_projective_plane: B is LINE of real_projective_plane}; then reconsider l = Line(Q,R) as Element of ProjectiveLines real_projective_plane; dual l in dual P & dual l in dual Q & dual l in dual R by A4,COLLSP:11,Th54,COLLSP:10; hence thesis; end; end; theorem Th57: for l being Element of ProjectiveLines real_projective_plane for P,Q,R being Point of real_projective_plane st P in l & Q in l & R in l holds P,Q,R are_collinear proof let l be Element of ProjectiveLines real_projective_plane; let P,Q,R be Point of real_projective_plane; assume A1: P in l & Q in l & R in l; l is LINE of real_projective_plane by INCPROJ:1; hence thesis by A1,COLLSP:16; end; theorem Th58: for l1,l2,l3 being Element of ProjectiveLines real_projective_plane st l1,l2,l3 are_concurrent holds dual l1,dual l2,dual l3 are_collinear proof let l1,l2,l3 be Element of ProjectiveLines real_projective_plane; assume l1,l2,l3 are_concurrent; then consider P be Point of real_projective_plane such that A1: P in l1 & P in l2 & P in l3; reconsider lP = dual P as Element of ProjectiveLines real_projective_plane; dual l1 in lP & dual l2 in lP & dual l3 in lP by A1,Th54; hence thesis by Th57; end; theorem Th59: for P,Q,R being Point of real_projective_plane holds P,Q,R are_collinear iff dual P,dual Q,dual R are_concurrent proof let P,Q,R be Point of real_projective_plane; thus P,Q,R are_collinear implies dual P, dual Q, dual R are_concurrent by Th56; assume dual P, dual Q, dual R are_concurrent; then dual dual P, dual dual Q, dual dual R are_collinear by Th58; then P, dual dual Q, dual dual R are_collinear by Th45; then P, Q, dual dual R are_collinear by Th45; hence thesis by Th45; end; theorem Th60: for l1,l2,l3 being Element of ProjectiveLines real_projective_plane holds l1,l2,l3 are_concurrent iff dual l1, dual l2, dual l3 are_collinear proof let l1,l2,l3 be Element of ProjectiveLines real_projective_plane; hereby assume l1,l2,l3 are_concurrent; then dual dual l1,l2,l3 are_concurrent by Th46; then dual dual l1,dual dual l2,l3 are_concurrent by Th46; then dual dual l1,dual dual l2,dual dual l3 are_concurrent by Th46; hence dual l1, dual l2, dual l3 are_collinear by Th59; end; assume dual l1, dual l2, dual l3 are_collinear; then dual dual l1, dual dual l2, dual dual l3 are_concurrent by Th59; then l1, dual dual l2, dual dual l3 are_concurrent by Th46; then l1, l2, dual dual l3 are_concurrent by Th46; hence thesis by Th46; end; begin :: Some converse theorems theorem real_projective_plane is reflexive & real_projective_plane is transitive & real_projective_plane is Vebleian & real_projective_plane is at_least_3rank & real_projective_plane is Fanoian & real_projective_plane is Desarguesian & real_projective_plane is Pappian & real_projective_plane is 2-dimensional; ::$N Converse reflexive theorem for l,m,n being Element of ProjectiveLines real_projective_plane holds l,m,l are_concurrent & l,l,m are_concurrent & l,m,m are_concurrent proof let l1,l2,l3 be Element of ProjectiveLines real_projective_plane; dual l1,dual l2,dual l1 are_collinear & dual l1,dual l1,dual l2 are_collinear & dual l1,dual l2,dual l2 are_collinear by ANPROJ_2:def 7; then dual dual l1,dual dual l2,dual dual l1 are_concurrent & dual dual l1,dual dual l1,dual dual l2 are_concurrent & dual dual l1,dual dual l2,dual dual l2 are_concurrent by Th59; then l1,dual dual l2,dual dual l1 are_concurrent & l1,dual dual l1,dual dual l2 are_concurrent & l1,dual dual l2,dual dual l2 are_concurrent by Th46; then l1,l2,dual dual l1 are_concurrent & l1,l1,dual dual l2 are_concurrent & l1,l2,dual dual l2 are_concurrent by Th46; hence thesis; end; ::$N Converse transitive theorem for l,m,n,n1,n2 being Element of ProjectiveLines real_projective_plane st l <> m & l,m,n are_concurrent & l,m,n1 are_concurrent & l,m,n2 are_concurrent holds n,n1,n2 are_concurrent proof let l,m,n,n1,n2 be Element of ProjectiveLines real_projective_plane; assume that A1: l <> m and A2: l,m,n are_concurrent and A3: l,m,n1 are_concurrent and A4: l,m,n2 are_concurrent; dual l <> dual m & dual l,dual m, dual n are_collinear & dual l,dual m, dual n1 are_collinear & dual l,dual m, dual n2 are_collinear by A1,A2,A3,A4,Th60,Th48; then dual dual n, dual dual n1, dual dual n2 are_concurrent by ANPROJ_2:def 8,Th59; then n, dual dual n1, dual dual n2 are_concurrent by Th46; then n, n1, dual dual n2 are_concurrent by Th46; hence thesis by Th46; end; ::$N Converse Vebliean theorem for l,l1,l2,n,n1 being Element of ProjectiveLines real_projective_plane st l,l1,n are_concurrent & l1,l2,n1 are_concurrent ex n2 being Element of ProjectiveLines real_projective_plane st l,l2,n2 are_concurrent & n,n1,n2 are_concurrent proof let l,l1,l2,n,n1 be Element of ProjectiveLines real_projective_plane; assume that A1: l,l1,n are_concurrent and A2: l1,l2,n1 are_concurrent; dual l, dual l1, dual n are_collinear & dual l1,dual l2, dual n1 are_collinear by A1,A2,Th60; then consider P be Point of real_projective_plane such that A3: dual l, dual l2, P are_collinear and A4: dual n, dual n1, P are_collinear by ANPROJ_2:def 9; take dual P; dual dual l, dual dual l2, dual P are_concurrent & dual dual n, dual dual n1, dual P are_concurrent by A3,A4,Th59; then l, dual dual l2, dual P are_concurrent & n, dual dual n1, dual P are_concurrent by Th46; hence thesis by Th46; end; ::$N Converse at_least_3rank theorem for l,m being Element of ProjectiveLines real_projective_plane holds ex n being Element of ProjectiveLines real_projective_plane st l <> n & m <> n & l,m,n are_concurrent proof let l,m be Element of ProjectiveLines real_projective_plane; consider r be Point of real_projective_plane such that A1: dual l <> r and A2: dual m <> r and A3: dual l, dual m, r are_collinear by ANPROJ_2:def 10; now thus l <> dual r & m <> dual r by Th45,A1,A2; dual dual l, dual dual m, dual r are_concurrent by A3,Th59; then l, dual dual m, dual r are_concurrent by Th46; hence l, m, dual r are_concurrent by Th46; end; hence thesis; end; ::$N Converse Fanoian theorem for l1,n2,m,n1,m1,l,n being Element of ProjectiveLines real_projective_plane holds (l1,n2,m are_concurrent & n1,m1,m are_concurrent & l1,n1,l are_concurrent & n2,m1,l are_concurrent & l1,m1,n are_concurrent & n2,n1,n are_concurrent & l,m,n are_concurrent implies (l1,n2,m1 are_concurrent or l1,n2,n1 are_concurrent or l1,n1,m1 are_concurrent or n2,n1,m1 are_concurrent)) proof let l1,n2,m,n1,m1,l,n be Element of ProjectiveLines real_projective_plane; assume that A1: l1,n2,m are_concurrent and A2: n1,m1,m are_concurrent and A3: l1,n1,l are_concurrent and A4: n2,m1,l are_concurrent and A5: l1,m1,n are_concurrent and A6: n2,n1,n are_concurrent and A7: l,m,n are_concurrent; dual l1,dual n2,dual m are_collinear & dual n1,dual m1,dual m are_collinear & dual l1,dual n1,dual l are_collinear & dual n2,dual m1,dual l are_collinear & dual l1,dual m1,dual n are_collinear & dual n2,dual n1,dual n are_collinear & dual l,dual m,dual n are_collinear by A1,A2,A3,A4,A5,A6,A7,Th60; then dual l1,dual n2,dual m1 are_collinear or dual l1,dual n2,dual n1 are_collinear or dual l1,dual n1,dual m1 are_collinear or dual n2,dual n1,dual m1 are_collinear by ANPROJ_2:def 11; hence thesis by Th60; end; ::$N Converse Desarguesian theorem for k,l1,l2,l3,m1,m2,m3,n1,n2,n3 being Element of ProjectiveLines real_projective_plane st k <> m1 & l1 <> m1 & k <> m2 & l2 <> m2 & k <> m3 & l3 <> m3 & not k,l1,l2 are_concurrent & not k,l1,l3 are_concurrent & not k,l2,l3 are_concurrent & l1,l2,n3 are_concurrent & m1,m2,n3 are_concurrent & l2,l3,n1 are_concurrent & m2,m3,n1 are_concurrent & l1,l3,n2 are_concurrent & m1,m3,n2 are_concurrent & k,l1,m1 are_concurrent & k,l2,m2 are_concurrent & k,l3,m3 are_concurrent holds n1,n2,n3 are_concurrent proof let k,l1,l2,l3,m1,m2,m3,n1,n2,n3 be Element of ProjectiveLines real_projective_plane; assume that A1: k <> m1 and A2: l1 <> m1 and A3: k <> m2 and A4: l2 <> m2 and A5: k <> m3 and A6: l3 <> m3 and A7: not k,l1,l2 are_concurrent and A8: not k,l1,l3 are_concurrent and A9: not k,l2,l3 are_concurrent and A10: l1,l2,n3 are_concurrent and A11: m1,m2,n3 are_concurrent and A12: l2,l3,n1 are_concurrent and A13: m2,m3,n1 are_concurrent and A14: l1,l3,n2 are_concurrent and A15: m1,m3,n2 are_concurrent and A16: k,l1,m1 are_concurrent and A17: k,l2,m2 are_concurrent and A18: k,l3,m3 are_concurrent; dual k <> dual m1 & dual l1 <> dual m1 & dual k <> dual m2 & dual l2 <> dual m2 & dual k <> dual m3 & dual l3 <> dual m3 & not dual k,dual l1,dual l2 are_collinear & not dual k,dual l1,dual l3 are_collinear & not dual k,dual l2,dual l3 are_collinear & dual l1,dual l2,dual n3 are_collinear & dual m1,dual m2,dual n3 are_collinear & dual l2,dual l3,dual n1 are_collinear & dual m2,dual m3,dual n1 are_collinear & dual l1,dual l3,dual n2 are_collinear & dual m1,dual m3,dual n2 are_collinear & dual k,dual l1,dual m1 are_collinear & dual k,dual l2,dual m2 are_collinear & dual k,dual l3,dual m3 are_collinear by A1,A2,A3,A4,A5,A6,Th48, A7,A8,A9,A10,A11,A12,A13,A14,A15,A16,A17,A18,Th60; then dual n1,dual n2, dual n3 are_collinear by ANPROJ_2:def 12; hence thesis by Th60; end; ::$N Converse Pappian theorem for k,l1,l2,l3,m1,m2,m3,n1,n2,n3 being Element of ProjectiveLines real_projective_plane st k <> l2 & k <> l3 & l2 <> l3 & l1 <> l2 & l1 <> l3 & k <> m2 & k <> m3 & m2 <> m3 & m1 <> m2 & m1 <> m3 & not k,l1,m1 are_concurrent & k,l1,l2 are_concurrent & k,l1,l3 are_concurrent & k,m1,m2 are_concurrent & k,m1,m3 are_concurrent & l1,m2,n3 are_concurrent & m1,l2,n3 are_concurrent & l1,m3,n2 are_concurrent & l3,m1,n2 are_concurrent & l2,m3,n1 are_concurrent & l3,m2,n1 are_concurrent holds n1,n2,n3 are_concurrent proof let k,l1,l2,l3,m1,m2,m3,n1,n2,n3 be Element of ProjectiveLines real_projective_plane; assume that A1: k <> l2 and A2: k <> l3 and A3: l2 <> l3 and A4: l1 <> l2 and A5: l1 <> l3 and A6: k <> m2 and A7: k <> m3 and A8: m2 <> m3 and A9: m1 <> m2 and A10: m1 <> m3 and A11: not k,l1,m1 are_concurrent and A12: k,l1,l2 are_concurrent and A13: k,l1,l3 are_concurrent and A14: k,m1,m2 are_concurrent and A15: k,m1,m3 are_concurrent and A16: l1,m2,n3 are_concurrent and A17: m1,l2,n3 are_concurrent and A18: l1,m3,n2 are_concurrent and A19: l3,m1,n2 are_concurrent and A20: l2,m3,n1 are_concurrent and A21: l3,m2,n1 are_concurrent; now thus dual k <> dual l2 & dual k <> dual l3 & dual l2 <> dual l3 by A1,A2,A3,Th48; thus dual l1 <> dual l2 & dual l1 <> dual l3 & dual k <> dual m2 by A4,A5,A6,Th48; thus dual k <> dual m3 & dual m2 <> dual m3 & dual m1 <> dual m2 & dual m1 <> dual m3 by A7,A8,A9,A10,Th48; thus not dual k,dual l1,dual m1 are_collinear & dual k,dual l1,dual l2 are_collinear & dual k,dual l1,dual l3 are_collinear & dual k,dual m1,dual m2 are_collinear & dual k,dual m1,dual m3 are_collinear & dual l1,dual m2,dual n3 are_collinear & dual m1,dual l2,dual n3 are_collinear & dual l1,dual m3,dual n2 are_collinear & dual l3,dual m1,dual n2 are_collinear & dual l2,dual m3,dual n1 are_collinear & dual l3,dual m2,dual n1 are_collinear by A11,A12,A13,A14,A15,A16,A17,A18,A19,A20,A21,Th60; end; then dual n1,dual n2, dual n3 are_collinear by ANPROJ_2:def 13; hence thesis by Th60; end; ::$N Converse 2_dimensional theorem for l,l1,m,m1 being Element of ProjectiveLines real_projective_plane holds ex n being Element of ProjectiveLines real_projective_plane st l,l1,n are_concurrent & m,m1,n are_concurrent proof let l,l1,m,m1 be Element of ProjectiveLines real_projective_plane; consider R be Point of real_projective_plane such that A1: dual l, dual l1, R are_collinear and A2: dual m, dual m1, R are_collinear by ANPROJ_2:def 14; dual dual l, dual dual l1, dual R are_concurrent & dual dual m, dual dual m1, dual R are_concurrent by A1,A2,Th59; then l, dual dual l1, dual R are_concurrent & m, dual dual m1, dual R are_concurrent by Th46; then l, l1, dual R are_concurrent & m, m1, dual R are_concurrent by Th46; hence thesis; end;