:: 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;