:: Analytical Ordered Affine Spaces :: by Henryk Oryszczyszyn and Krzysztof Pra\.zmowski environ vocabularies NUMBERS, RLVECT_1, REAL_1, CARD_1, ARYTM_3, RELAT_1, ARYTM_1, SUPINF_2, STRUCT_0, ZFMISC_1, XBOOLE_0, SUBSET_1, ANALOAF; notations TARSKI, XBOOLE_0, ZFMISC_1, ORDINAL1, DOMAIN_1, XXREAL_0, XCMPLX_0, XREAL_0, REAL_1, RELSET_1, NUMBERS, STRUCT_0, RLVECT_1; constructors XXREAL_0, REAL_1, MEMBERED, DOMAIN_1, RLVECT_1; registrations SUBSET_1, RELSET_1, XXREAL_0, STRUCT_0, ZFMISC_1, XREAL_0, ORDINAL1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; equalities RLVECT_1; theorems RLVECT_1, RELAT_1, FUNCSDOM, RLSUB_2, XCMPLX_0, XCMPLX_1, XREAL_1, STRUCT_0, XTUPLE_0; schemes RELSET_1; begin reserve V for RealLinearSpace; reserve p,q,u,v,w,y for VECTOR of V; reserve a,b,c,d for Real; definition let V; let u,v,w,y; pred u,v // w,y means u=v or w=y or ex a,b st 00 & a*u=v implies u=a"*v proof assume that A1: a<>0 and A2: a*u=v; thus u=1*u by RLVECT_1:def 8 .=(a"*a)*u by A1,XCMPLX_0:def 7 .=a"*v by A2,RLVECT_1:def 7; end; theorem Th6: (a<>0 & a*u=v implies u=a"*v) & (a<>0 & u=a"*v implies a*u=v) proof now assume a<>0 & u=a"*v; hence v=(a")"*u by Th5,XCMPLX_1:202 .=a*u; end; hence thesis by Th5; end; theorem u,v // w,y & u<>v & w<>y implies ex a,b st a*(v-u)=b*(y-w) & 0v; then consider a,b such that A3: a*(v-u)=b*(u-v) and A4: 0q & p,q // u,v & p,q // w,y implies u,v // w,y proof assume that A1: p<>q and A2: p,q // u,v and A3: p,q // w,y; now assume that A4: u<>v and A5: w<>y; consider a,b such that A6: a*(q-p)=b*(v-u) and A7: 0v & w<>y; then consider a,b such that A2: a*(v-u)=b*(y-w) and A3: 0v & v<>w; then consider a,b such that A2: a*(v-u)=b*(w-v) and A3: 0v & u<>w; then consider a,b such that A2: a*(v-u)=b*(w-u) and A3: 0b; then 0q) implies for u,v,w ex y st u,v // w,y & u,w // v ,y & v<>y proof given p,q such that A1: p<>q; let u,v,w; A2: now assume A3: u<>w; take y=(v+w)-u; A4: now assume v=y; then v=v+(w-u) by RLVECT_1:def 3; then w-u=0.V by RLVECT_1:9; hence contradiction by A3,RLVECT_1:21; end; u,v // w,y & u,w // v,y by Th16; hence thesis by A4; end; now assume A5: u=w; A6: now assume u=v; then A7: u,v // w,p & u,v // w,q; A8: v<>p or v<>q by A1; u,w // v,p & u,w // v,q by A5; hence thesis by A8,A7; end; u,v // w,u & u,w // v,u by A5; hence thesis by A6; end; hence thesis by A2; end; theorem Th18: p<>v & v,p // p,w implies ex y st u,p // p,y & u,v // w,y proof assume A1: p<>v & v,p // p,w; A2: now assume p<>w; then consider a,b such that A3: a*(p-v)=b*(w-p) and A4: 0v & u<>0.V & v<>0.V proof assume A1: for a,b st a*u + b*v=0.V holds a=0 & b=0; thus u<>v proof assume u=v; then u - v = 0.V by RLVECT_1:15; then 1*u + (-v) = 0.V by RLVECT_1:def 8; then 1*u + ((-jj)*v) = 0.V by RLVECT_1:16; hence contradiction by A1; end; thus u<>0.V proof assume u=0.V; then 1*u = 0.V by RLVECT_1:10; then 1*u + 0.V = 0.V by RLVECT_1:4; then jj*u + 0*v =0.V by RLVECT_1:10; hence contradiction by A1; end; thus v<>0.V proof assume v=0.V; then 1*v = 0.V by RLVECT_1:10; then 0.V + 1*v = 0.V by RLVECT_1:4; then 0*u + jj*v =0.V by RLVECT_1:10; hence contradiction by A1; end; end; theorem Th20: (ex u,v st (for a,b st a*u + b*v=0.V holds a=0 & b=0)) implies ex u,v,w,y st not u,v // w,y & not u,v // y,w proof given u,v such that A1: for a,b st a*u + b*v=0.V holds a=0 & b=0; A2: u<>0.V & v<>0.V by A1,Th19; A3: not 0.V,u // v,0.V proof A4: now given a,b such that A5: 00 or b<>0) implies u,v // w,y or u,v // y,w proof assume that A1: a*(v-u) = b*(w-y) and A2: a<>0 or b<>0; A3: now assume A4: b=0; then 0.V = a*(v-u) by A1,RLVECT_1:10; then v-u = 0.V by A2,A4,RLVECT_1:11; then u=v by RLVECT_1:21; hence u,v // w,y; end; A5: now A6: now A7: a*(v-u) = -(-(b*(w-y))) by A1,RLVECT_1:17 .= -(b*(-(w-y))) by RLVECT_1:25 .= -(b*(y-w)) by RLVECT_1:33 .= b*(-(y-w)) by RLVECT_1:25 .= (-b)*(y-w) by RLVECT_1:24; assume that A8: 00 & b<>0; hence thesis by A1,A14,A10,A6; end; now assume A16: a=0; then 0.V = b*(w-y) by A1,RLVECT_1:10; then w-y = 0.V by A2,A16,RLVECT_1:11; then w=y by RLVECT_1:21; hence u,v // w,y; end; hence thesis by A3,A5; end; theorem Th21: (ex p,q st (for w ex a,b st a*p + b*q=w)) implies for u,v,w,y st not u,v // w,y & not u,v // y,w ex z being VECTOR of V st (u,v // u,z or u,v // z,u) & (w,y // w,z or w,y // z,w) proof given p,q such that A1: for w ex a,b st a*p + b*q=w; let u,v,w,y such that A2: not u,v // w,y and A3: not u,v // y,w; consider r1,s1 being Real such that A4: r1*p + s1*q = v-u by A1; consider r2,s2 being Real such that A5: r2*p + s2*q = y-w by A1; set r = r1*s2 - r2*s1; A6: now assume A7: r = 0; A8: now assume that A9: r1<>0 and A10: r2=0; s2<>0 proof assume s2=0; then y-w = 0.V + 0*q by A5,A10,RLVECT_1:10 .= 0.V + 0.V by RLVECT_1:10 .= 0.V by RLVECT_1:4; then y=w by RLVECT_1:21; hence contradiction by A2; end; hence contradiction by A7,A9,A10,XCMPLX_1:6; end; A11: now assume A12: r1=0; A13: s1<>0 proof assume s1=0; then v-u = 0.V + 0*q by A4,A12,RLVECT_1:10 .= 0.V + 0.V by RLVECT_1:10 .= 0.V by RLVECT_1:4; then u=v by RLVECT_1:21; hence contradiction by A2; end; then A14: r2=0 by A7,A12,XCMPLX_1:6; A15: s2<>0 proof assume s2=0; then y-w = 0.V + 0*q by A5,A14,RLVECT_1:10 .= 0.V + 0.V by RLVECT_1:10 .= 0.V by RLVECT_1:4; then y=w by RLVECT_1:21; hence contradiction by A2; end; y-w = 0.V + s2*q by A5,A14,RLVECT_1:10 .= s2*q by RLVECT_1:4; then A16: (s2)"*(y-w) = ((s2)"*s2)*q by RLVECT_1:def 7 .= 1*q by A15,XCMPLX_0:def 7 .= q by RLVECT_1:def 8; v-u = 0.V + s1*q by A4,A12,RLVECT_1:10 .= s1*q by RLVECT_1:4; then A17: (s1)"*(v-u) = ((s1)"*s1)*q by RLVECT_1:def 7 .= 1*q by A13,XCMPLX_0:def 7 .= q by RLVECT_1:def 8; s1"<>0 by A13,XCMPLX_1:202; hence contradiction by A2,A3,A17,A16,Lm1; end; A18: now assume that A19: r1<>0 and A20: r2<>0 and A21: s1 = 0; v-u = r1*p + 0.V by A4,A21,RLVECT_1:10 .= r1*p by RLVECT_1:4; then A22: (r1)"*(v-u) = ((r1)"*r1)*p by RLVECT_1:def 7 .= 1*p by A19,XCMPLX_0:def 7 .= p by RLVECT_1:def 8; s2 = 0 by A7,A19,A21,XCMPLX_1:6; then y-w = r2*p + 0.V by A5,RLVECT_1:10 .= r2*p by RLVECT_1:4; then A23: (r2)"*(y-w) = ((r2)"*r2)*p by RLVECT_1:def 7 .= 1*p by A20,XCMPLX_0:def 7 .= p by RLVECT_1:def 8; r1"<>0 by A19,XCMPLX_1:202; hence contradiction by A2,A3,A22,A23,Lm1; end; now assume that A24: r1<>0 and r2<>0 and s1<>0 and s2<>0; r2*(v-u) = r2*(r1*p) + r2*(s1*q) by A4,RLVECT_1:def 5 .=(r2*r1)*p + r2*(s1*q) by RLVECT_1:def 7 .= (r1*r2)*p + (r1*s2)*q by A7,RLVECT_1:def 7 .= r1*(r2*p) + (r1*s2)*q by RLVECT_1:def 7 .= r1*(r2*p) + r1*(s2*q) by RLVECT_1:def 7 .= r1*(y-w) by A5,RLVECT_1:def 5; hence contradiction by A2,A3,A24,Lm1; end; hence contradiction by A7,A11,A8,A18,XCMPLX_1:6; end; consider r3,s3 being Real such that A25: r3*p + s3*q = u-w by A1; set a= r2*s3 - r3*s2, b= r1*s3 - r3*s1; A26: b*r2 = r1*a + r3*r; set z = u + (r"*a)*(v-u); A27: r*(z-u) = r*z - r*u by RLVECT_1:34 .= r*u + r*((r"*a)*(v-u)) - r*u by RLVECT_1:def 5 .= r*u + (r*(r"*a))*(v-u) - r*u by RLVECT_1:def 7 .= r*u + ((r*r")*a)*(v-u) - r*u .= r*u + (1*a)*(v-u) - r*u by A6,XCMPLX_0:def 7 .= a*(v-u) + (r*u - r*u) by RLVECT_1:def 3 .= a*(v-u) + 0.V by RLVECT_1:15 .= a*(v-u) by RLVECT_1:4; A28: r*(z-w) = r*z - r*w by RLVECT_1:34 .= r*u + r*((r"*a)*(v-u)) - r*w by RLVECT_1:def 5 .= r*u + (r*(r"*a))*(v-u) - r*w by RLVECT_1:def 7 .= r*u + ((r*r")*a)*(v-u) - r*w .= r*u + (1*a)*(v-u) - r*w by A6,XCMPLX_0:def 7 .= a*(v-u) + (r*u - r*w) by RLVECT_1:def 3 .= a*(r1*p + s1*q) + r*(r3*p + s3*q) by A4,A25,RLVECT_1:34 .= a*(r1*p) + a*(s1*q) + r*(r3*p + s3*q) by RLVECT_1:def 5 .= a*(r1*p) + a*(s1*q) + (r*(r3*p) + r*(s3*q)) by RLVECT_1:def 5 .= (a*r1)*p + a*(s1*q) + (r*(r3*p) + r*(s3*q)) by RLVECT_1:def 7 .= (a*r1)*p + (a*s1)*q + (r*(r3*p) + r*(s3*q)) by RLVECT_1:def 7 .= (a*r1)*p + (a*s1)*q + ((r*r3)*p + r*(s3*q)) by RLVECT_1:def 7 .= (a*r1)*p + (a*s1)*q + ((r*s3)*q + (r*r3)*p) by RLVECT_1:def 7 .= (a*r1)*p + (a*s1)*q + (r*s3)*q + (r*r3)*p by RLVECT_1:def 3 .= ((a*s1)*q + (r*s3)*q) + (a*r1)*p + (r*r3)*p by RLVECT_1:def 3 .= ((a*s1)*q + (r*s3)*q) + ((a*r1)*p + (r*r3)*p) by RLVECT_1:def 3 .= (a*s1 + r*s3)*q + ((a*r1)*p + (r*r3)*p) by RLVECT_1:def 6 .= (b*s2)*q + (b*r2)*p by A26,RLVECT_1:def 6 .= b*(s2*q) + (b*r2)*p by RLVECT_1:def 7 .= b*(s2*q) + b*(r2*p) by RLVECT_1:def 7 .= b*(y-w) by A5,RLVECT_1:def 5; A29: b*s2 = s1*a + s3*r; per cases; suppose that A30: a=0 and A31: b<>0; r*(z-u)=0.V by A27,A30,RLVECT_1:10; then z-u=0.V by A6,RLVECT_1:11; then z=u by RLVECT_1:21; then A32: u,v // u,z; w,y // w,z or w,y // z,w by A28,A31,Lm1; hence thesis by A32; end; suppose a=0 & b=0; then r3=0 & s3=0 by A6,A26,A29,XCMPLX_1:6; then 0.V + 0*q = u-w by A25,RLVECT_1:10; then 0.V + 0.V = u-w by RLVECT_1:10; then 0.V=u-w by RLVECT_1:4; then u=w by RLVECT_1:21; then A33: w,y // w,u; u,v // u,u; hence thesis by A33; end; suppose that A34: a<>0 and A35: b=0; r*(z-w)=0.V by A28,A35,RLVECT_1:10; then z-w=0.V by A6,RLVECT_1:11; then z=w by RLVECT_1:21; then A36: w,y // w,z; u,v // u,z or u,v // z,u by A27,A34,Lm1; hence thesis by A36; end; suppose that A37: a<>0 and A38: b<>0; A39: w,y // w,z or w,y // z,w by A28,A38,Lm1; u,v // u,z or u,v // z,u by A27,A37,Lm1; hence thesis by A39; end; end; definition struct(1-sorted) AffinStruct (#carrier -> set, CONGR -> Relation of [:the carrier,the carrier:]#); end; registration cluster non trivial strict for AffinStruct; existence proof set A = the non trivial set, R = the Relation of [:A,A:]; take AffinStruct(#A,R#); thus thesis; end; end; reserve AS for non empty AffinStruct; reserve a,b,c,d for Element of AS; reserve x,z for object; definition let AS,a,b,c,d; pred a,b // c,d means [[a,b],[c,d]] in the CONGR of AS; end; definition let V; func DirPar(V) -> Relation of [:the carrier of V,the carrier of V:] means :Def3: [x,z] in it iff ex u,v,w,y st x=[u,v] & z=[w,y] & u,v // w,y; existence proof defpred P[object,object] means ex u,v,w,y st $1=[u,v] & $2=[w,y] & u,v // w,y; set VV = [:the carrier of V,the carrier of V:]; consider P being Relation of VV,VV such that A1: for x,z being object holds [x,z] in P iff x in VV & z in VV & P[x,z] from RELSET_1:sch 1; take P; let x,z; thus [x,z] in P implies ex u,v,w,y st x=[u,v] & z=[w,y] & u,v // w,y by A1; assume ex u,v,w,y st x=[u,v] & z=[w,y] & u,v // w,y; hence thesis by A1; end; uniqueness proof let P,Q be Relation of [:the carrier of V,the carrier of V:] such that A2: [x,z] in P iff ex u,v,w,y st x=[u,v] & z=[w,y] & u,v // w,y and A3: [x,z] in Q iff ex u,v,w,y st x=[u,v] & z=[w,y] & u,v // w,y; for x,z being object holds [x,z] in P iff [x,z] in Q proof let x,z be object; [x,z] in P iff ex u,v,w,y st x=[u,v] & z=[w,y] & u,v // w,y by A2; hence thesis by A3; end; hence thesis by RELAT_1:def 2; end; end; theorem Th22: [[u,v],[w,y]] in DirPar(V) iff u,v // w,y proof thus [[u,v],[w,y]] in DirPar(V) implies u,v // w,y proof assume [[u,v],[w,y]] in DirPar(V); then consider u9,v9,w9,y9 being VECTOR of V such that A1: [u,v]=[u9,v9] and A2: [w,y]=[w9,y9] and A3: u9,v9 // w9,y9 by Def3; A4: w = w9 by A2,XTUPLE_0:1; u = u9 & v = v9 by A1,XTUPLE_0:1; hence thesis by A2,A3,A4,XTUPLE_0:1; end; thus thesis by Def3; end; definition let V; func OASpace(V) -> strict AffinStruct equals AffinStruct (#the carrier of V, DirPar(V)#); correctness; end; registration let V; cluster OASpace V -> non empty; coherence; end; theorem Th23: (ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0 ) implies (ex a,b being Element of OASpace(V) st a<>b) & (for a,b,c,d,p,q,r,s being Element of OASpace(V) holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c, b)) & (ex a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b // d, c) & (for a,b,c being Element of OASpace(V) ex d being Element of OASpace(V) st a,b // c,d & a,c // b,d & b<>d) & for p,a,b,c being Element of OASpace(V) st p <>b & b,p // p,c ex d being Element of OASpace(V) st a,p // p,d & a,b // c,d proof given u,v such that A1: for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0; set S = OASpace(V); A2: u<>v by A1,Th19; hence ex a,b being Element of S st a<>b; thus for a,b,c,d,p,q,r,s being Element of S holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c, d implies b,a // d,c) & (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a ,b // b,c or a,c // c,b) proof let a,b,c,d,p,q,r,s be Element of S; reconsider a9=a,b9=b,c9=c,d9=d,p9=p,q9=q,r9=r,s9=s as Element of V; a9,b9 // c9,c9; hence [[a,b],[c,c]] in the CONGR of S by Def3; thus a,b // b,a implies a=b by Th22,Th10; thus a<>b & a,b // p,q & a,b // r,s implies p,q // r,s proof assume that A3: a<>b and A4: [[a,b],[p,q]] in the CONGR of S & [[a,b],[r,s]] in the CONGR of S; a9,b9 // p9,q9 & a9,b9 // r9,s9 by A4,Th22; then p9,q9 // r9,s9 by A3,Th11; then [[p,q],[r,s]] in the CONGR of S by Th22; hence thesis; end; thus a,b // c,d implies b,a // d,c proof assume [[a,b],[c,d]] in the CONGR of S; then a9,b9 // c9,d9 by Th22; then b9,a9 // d9,c9 by Th12; then [[b,a],[d,c]] in the CONGR of S by Th22; hence thesis; end; thus a,b // b,c implies a,b // a,c proof assume [[a,b],[b,c]] in the CONGR of S; then a9,b9 // b9,c9 by Th22; then a9,b9 // a9,c9 by Th13; then [[a,b],[a,c]] in the CONGR of S by Th22; hence thesis; end; thus a,b // a,c implies a,b // b,c or a,c // c,b proof assume [[a,b],[a,c]] in the CONGR of S; then a9,b9 // a9,c9 by Th22; then a9,b9 // b9,c9 or a9,c9 // c9,b9 by Th14; then [[a,b],[b,c]] in the CONGR of S or [[a,c],[c,b]] in the CONGR of S by Th22; hence thesis; end; end; thus ex a,b,c,d being Element of S st not a,b // c,d & not a,b // d,c proof consider a9,b9,c9,d9 being VECTOR of V such that A5: not a9,b9 // c9,d9 and A6: not a9,b9 // d9,c9 by A1,Th20; reconsider a=a9,b=b9,c = c9,d=d9 as Element of S; not [[a,b],[d,c]] in the CONGR of S by A6,Th22; then A7: not a,b // d,c; not [[a,b],[c,d]] in the CONGR of S by A5,Th22; then not a,b // c,d; hence thesis by A7; end; thus for a,b,c being Element of S ex d being Element of S st a,b // c,d & a, c // b,d & b<>d proof let a,b,c be Element of S; reconsider a9=a,b9=b,c9=c as Element of V; consider d9 being VECTOR of V such that A8: a9,b9 // c9,d9 and A9: a9,c9 // b9,d9 and A10: b9<>d9 by A2,Th17; reconsider d=d9 as Element of S; [[a,c],[b,d]] in the CONGR of S by A9,Th22; then A11: a,c // b,d; [[a,b],[c,d]] in the CONGR of S by A8,Th22; then a,b // c,d; hence thesis by A10,A11; end; thus for p,a,b,c being Element of S st p<>b & b,p // p,c holds ex d being Element of S st a,p // p,d & a,b // c,d proof let p,a,b,c be Element of S; assume that A12: p<>b and A13: [[b,p],[p,c]] in the CONGR of S; reconsider p9=p,a9=a,b9=b,c9=c as Element of V; b9,p9 // p9,c9 by A13,Th22; then consider d9 being VECTOR of V such that A14: a9,p9 // p9,d9 and A15: a9,b9 // c9,d9 by A12,Th18; reconsider d=d9 as Element of S; [[a,b],[c,d]] in the CONGR of S by A15,Th22; then A16: a,b // c,d; [[a,p],[p,d]] in the CONGR of S by A14,Th22; then a,p // p,d; hence thesis by A16; end; end; theorem Th24: (ex p,q being VECTOR of V st (for w being VECTOR of V ex a,b being Real st a*p + b*q=w)) implies for a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b // d,c ex t being Element of OASpace(V) st (a,b // a,t or a,b // t,a) & (c,d // c,t or c,d // t,c) proof assume A1: ex p,q being VECTOR of V st for w being VECTOR of V ex a,b being Real st a*p + b*q=w; set S = OASpace(V); let a,b,c,d be Element of OASpace(V); reconsider a9=a,b9=b,c9 = c,d9=d as Element of V; assume ( not [[a,b],[c,d]] in the CONGR of S)& not [[a,b],[d,c]] in the CONGR of S; then ( not a9,b9 // c9,d9)& not a9,b9 // d9,c9 by Th22; then consider t9 being VECTOR of V such that A2: a9,b9 // a9,t9 or a9,b9 // t9,a9 and A3: c9,d9 // c9,t9 or c9,d9 // t9,c9 by A1,Th21; reconsider t=t9 as Element of S; [[c,d],[c,t]] in the CONGR of S or [[c,d],[t,c]] in the CONGR of S by A3,Th22 ; then A4: c,d // c,t or c,d // t,c; [[a,b],[a,t]] in the CONGR of S or [[a,b],[t,a]] in the CONGR of S by A2,Th22 ; then a,b // a,t or a,b // t,a; hence thesis by A4; end; definition let IT be non empty AffinStruct; attr IT is OAffinSpace-like means :Def5: (for a,b,c,d,p,q,r,s being Element of IT holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b)) & (ex a,b,c,d being Element of IT st not a,b // c,d & not a,b // d,c) & (for a,b,c being Element of IT ex d being Element of IT st a,b // c,d & a,c // b,d & b<>d) & for p,a,b,c being Element of IT st p<>b & b,p // p,c ex d being Element of IT st a, p // p,d & a,b // c,d; end; registration cluster strict OAffinSpace-like for non trivial AffinStruct; existence proof consider V,u,v such that A1: for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0 and for w ex a,b being Real st w = a*u + b*v by FUNCSDOM:23; A2: ( ex a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b // d,c)& for a,b,c being Element of OASpace(V) ex d being Element of OASpace(V ) st a,b // c,d & a,c // b,d & b<>d by A1,Th23; A3: for p,a,b,c being Element of OASpace(V) st p<>b & b,p // p,c ex d being Element of OASpace(V) st a,p // p,d & a,b // c,d by A1,Th23; ( ex a,b being Element of OASpace(V) st a<>b)& for a,b,c,d,p,q,r,s being Element of OASpace(V) holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a, b // c,d implies b,a // d,c ) & (a,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b) by A1,Th23; then OASpace(V) is non trivial OAffinSpace-like by A2,A3, STRUCT_0:def 10; hence thesis; end; end; definition mode OAffinSpace is OAffinSpace-like non trivial AffinStruct; end; theorem (ex a,b being Element of AS st a<>b) & (for a,b,c,d,p,q,r,s being Element of AS holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b, c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b)) & (ex a, b,c,d being Element of AS st not a,b // c,d & not a,b // d,c) & (for a,b,c being Element of AS ex d being Element of AS st a,b // c,d & a,c // b,d & b<>d) & (for p,a,b,c being Element of AS st p<>b & b,p // p,c ex d being Element of AS st a,p // p,d & a,b // c,d) iff AS is OAffinSpace by Def5,STRUCT_0:def 10; theorem Th26: (ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0 ) implies OASpace(V) is OAffinSpace proof assume A1: ex u,v st for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0; then A2: ( ex a,b,c,d being Element of OASpace(V) st not a,b // c,d & not a,b // d,c)& for a,b,c being Element of OASpace(V) ex d being Element of OASpace(V) st a,b // c,d & a,c // b,d & b<>d by Th23; A3: for p,a,b,c being Element of OASpace(V) st p<>b & b,p // p,c ex d being Element of OASpace(V) st a,p // p,d & a,b // c,d by A1,Th23; ( ex a,b being Element of OASpace(V) st a<>b)& for a,b,c,d,p,q,r,s being Element of OASpace(V) holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a, b // c,d implies b,a // d,c) & (a ,b // b,c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b) by A1,Th23; hence thesis by A2,A3,Def5,STRUCT_0:def 10; end; definition let IT be OAffinSpace; attr IT is 2-dimensional means :Def6: for a,b,c,d being Element of IT st not a,b // c,d & not a,b // d,c holds ex p being Element of IT st (a,b // a,p or a, b // p,a) & (c,d // c,p or c,d // p,c); end; registration cluster strict 2-dimensional for OAffinSpace; existence proof consider V such that A1: ex u,v st (for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0) & for w ex a,b being Real st w = a*u + b*v by FUNCSDOM:23; reconsider S = OASpace(V) as OAffinSpace by A1,Th26; for a,b,c,d being Element of S st not a,b // c,d & not a,b // d,c holds ex p being Element of S st (a,b // a,p or a,b // p,a) & (c,d // c,p or c, d // p,c) by A1,Th24; then S is 2-dimensional; hence thesis; end; end; definition mode OAffinPlane is 2-dimensional OAffinSpace; end; theorem (ex a,b being Element of AS st a<>b) & (for a,b,c,d,p,q,r,s being Element of AS holds a,b // c,c & (a,b // b,a implies a=b) & (a<>b & a,b // p,q & a,b // r,s implies p,q // r,s) & (a,b // c,d implies b,a // d,c) & (a,b // b, c implies a,b // a,c) & (a,b // a,c implies a,b // b,c or a,c // c,b)) & (ex a, b,c,d being Element of AS st not a,b // c,d & not a,b // d,c) & (for a,b,c being Element of AS ex d being Element of AS st a,b // c,d & a,c // b,d & b<>d) & (for p,a,b,c being Element of AS st p<>b & b,p // p,c ex d being Element of AS st a,p // p,d & a,b // c,d) & (for a,b,c,d being Element of AS st not a,b // c,d & not a,b // d,c holds ex p being Element of AS st (a,b // a,p or a,b // p, a) & (c,d // c,p or c,d // p,c)) iff AS is OAffinPlane by Def5,Def6, STRUCT_0:def 10; theorem (ex u,v st (for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0) & (for w ex a,b being Real st w = a*u + b*v)) implies OASpace(V) is OAffinPlane proof set S=OASpace(V); assume A1: ex u,v st (for a,b being Real st a*u + b*v = 0.V holds a=0 & b=0) & for w ex a,b being Real st w = a*u + b*v; then for a,b,c,d being Element of S st not a,b // c,d & not a,b // d,c holds ex p being Element of S st (a,b // a,p or a,b // p,a) & (c,d // c,p or c,d // p ,c) by Th24; hence thesis by A1,Def6,Th26; end;