Copyright (c) 1991 Association of Mizar Users
environ vocabulary RLVECT_1, ARYTM_1, ANALMETR, FUNCT_3, ANALOAF, SYMSP_1, RELAT_1, ANALORT; notation TARSKI, XBOOLE_0, ZFMISC_1, REAL_1, RELSET_1, RLVECT_1, STRUCT_0, ANALOAF, ANALMETR, GEOMTRAP; constructors REAL_1, ANALMETR, GEOMTRAP, TDGROUP, DOMAIN_1, XREAL_0, MEMBERED, XBOOLE_0; clusters TDGROUP, RELSET_1, STRUCT_0, MEMBERED, ZFMISC_1, XBOOLE_0; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions STRUCT_0; theorems GEOMTRAP, RELAT_1, REAL_1, RLVECT_1, ANALOAF, ANALMETR, ZFMISC_1, XCMPLX_0, XCMPLX_1; schemes RELSET_1; begin definition let V be Abelian (non empty LoopStr), v,w be Element of V; redefine func v + w; commutativity by RLVECT_1:def 5; end; reserve V for RealLinearSpace, u,u1,u2,v,v1,v2,w,w1,x,y for VECTOR of V, a,a1,a2,b,b1,b2,c1,c2,n,k1,k2 for Real; Lm1: v1 = b1*x + b2*y & v2 = c1*x + c2*y implies v1 + v2 = (b1 + c1)*x + (b2 + c2)*y & v1 - v2 = (b1 - c1)*x + (b2 - c2)*y proof assume that A1: v1 = b1*x + b2*y and A2: v2 = c1*x + c2*y; thus v1 + v2 = ((b1*x + b2*y) + c1*x) + c2*y by A1,A2,RLVECT_1:def 6 .= ((b1*x + c1*x) + b2*y) + c2*y by RLVECT_1:def 6 .= ((b1 + c1)*x + b2*y) + c2*y by RLVECT_1:def 9 .= (b1 + c1)*x + (b2*y + c2*y) by RLVECT_1:def 6 .= (b1 + c1)*x + (b2 + c2)*y by RLVECT_1:def 9; thus v1 - v2 = (b1*x + b2*y)+(-(c1*x + c2*y)) by A1,A2,RLVECT_1:def 11 .= (b1*x + b2*y)+(-(c1*x) + -(c2*y)) by RLVECT_1:45 .= (b1*x + b2*y)+(c1*(-x) + -(c2*y)) by RLVECT_1:39 .= (b1*x + b2*y)+(c1*(-x) + c2*(-y)) by RLVECT_1:39 .= (b1*x + b2*y)+((-c1)*x + c2*(-y)) by RLVECT_1:38 .= (b1*x + b2*y)+((-c1)*x + (-c2)*y) by RLVECT_1:38 .= ((b1*x + b2*y) + (-c1)*x) + (-c2)*y by RLVECT_1:def 6 .= ((b1*x + (-c1)*x) + b2*y) + (-c2)*y by RLVECT_1:def 6 .= ((b1 + (-c1))*x + b2*y) + (-c2)*y by RLVECT_1:def 9 .= (b1 + (-c1))*x + (b2*y + (-c2)*y) by RLVECT_1:def 6 .= (b1 + (-c1))*x + (b2 + (-c2))*y by RLVECT_1:def 9 .= (b1 - c1)*x + (b2 + (-c2))*y by XCMPLX_0:def 8 .= (b1 - c1)*x + (b2 - c2)*y by XCMPLX_0:def 8; end; Lm2: v = b1*x + b2*y implies a*v = (a*b1)*x + (a*b2)*y proof assume v= b1*x + b2*y; hence a*v = a*(b1*x) + a*(b2*y) by RLVECT_1:def 9 .= (a*b1)*x + a*(b2*y) by RLVECT_1:def 9 .= (a*b1)*x + (a*b2)*y by RLVECT_1:def 9; end; Lm3: Gen x,y & a1*x + a2*y = b1*x + b2*y implies a1=b1 & a2=b2 proof assume that A1: Gen x,y and A2: a1*x+a2*y=b1*x+b2*y; 0.V = (a1*x+a2*y)-(b1*x+b2*y) by A2,RLVECT_1:28 .= (a1-b1)*x+(a2-b2)*y by Lm1; then a1-b1=0 & a2-b2=0 by A1,ANALMETR:def 1; then -b1 + a1 =0 & -b2 + a2 = 0 by XCMPLX_0:def 8; then a1=-(-b1) & a2=-(-b2) by XCMPLX_0:def 6; hence thesis; end; Lm4:Gen x,y implies x<>0.V & y<>0.V proof assume A1:Gen x,y; A2:x<>0.V proof assume A3:x=0.V; consider a,b such that A4:a=1 and A5:b=0; a*x+b*y=0.V+0*y by A3,A5,RLVECT_1:23 .=0.V+0.V by RLVECT_1:23 .=0.V by RLVECT_1:10; hence contradiction by A1,A4,ANALMETR:def 1; end; y<>0.V proof assume A6:y=0.V; consider a,b such that A7:a=0 and A8:b=1; a*x+b*y=0.V+1*0.V by A6,A7,A8,RLVECT_1:23 .=0.V+0.V by RLVECT_1:23 .=0.V by RLVECT_1:10; hence thesis by A1,A8,ANALMETR:def 1;end; hence thesis by A2;end; Lm5: Gen x,y implies u = pr1(x,y,u)*x + pr2(x,y,u)*y proof assume A1: Gen x,y; then ex b st u = (pr1(x,y,u))*x + b*y by GEOMTRAP:def 4; hence thesis by A1,GEOMTRAP:def 5; end; Lm6: (Gen x,y & u=k1*x+k2*y) implies k1=pr1(x,y,u) & k2=pr2(x,y,u) proof assume A1: Gen x,y & u=k1*x+k2*y; then u = pr1(x,y,u)*x + pr2(x,y,u)*y by Lm5; hence thesis by A1,Lm3; end; Lm7: Gen x,y implies ( pr1(x,y,u+v) = pr1(x,y,u)+pr1(x,y,v) & pr2(x,y,u+v) = pr2(x,y,u)+pr2(x,y,v) & pr1(x,y,u-v) = pr1(x,y,u)-pr1(x,y,v) & pr2(x,y,u-v) = pr2(x,y,u)-pr2(x,y,v) & pr1(x,y,a*u) = a*pr1(x,y,u) & pr2(x,y,a*u) = a*pr2(x,y,u)) proof assume A1: Gen x,y; set p1u = pr1(x,y,u), p2u = pr2(x,y,u), p1v = pr1(x,y,v), p2v = pr2(x,y,v); A2: u = p1u*x + p2u*y & v = p1v*x + p2v*y by A1,Lm5; then u + v = (p1u*x + p2u*y + p1v*x) + p2v*y by RLVECT_1:def 6 .= ((p1u*x + p1v*x) + p2u*y) + p2v*y by RLVECT_1:def 6 .= (p1u*x + p1v*x) + (p2u*y + p2v*y) by RLVECT_1:def 6 .= (p1u + p1v)*x + (p2u*y + p2v*y) by RLVECT_1:def 9 .= (p1u + p1v)*x + (p2u + p2v)*y by RLVECT_1:def 9; hence pr1(x,y,u+v) = p1u + p1v & pr2(x,y,u+v) = p2u + p2v by A1,Lm6; u - v = (p1u - p1v)*x + (p2u - p2v)*y by A2,Lm1; hence pr1(x,y,u-v) = p1u - p1v & pr2(x,y,u-v) = p2u - p2v by A1,Lm6; a*u = (a*p1u)*x + (a*p2u)*y by A2,Lm2; hence pr1(x,y,a*u) = a*pr1(x,y,u) & pr2(x,y,a*u) = a*pr2(x,y,u) by A1,Lm6; end; definition let V,x,y; let u; func Ortm(x,y,u) -> VECTOR of V equals :Def1: pr1(x,y,u)*x + (-pr2(x,y,u))*y; correctness; end; theorem Th1: Gen x,y implies Ortm(x,y,u+v)=Ortm(x,y,u)+Ortm(x,y,v) proof assume A1: Gen x,y; thus Ortm(x,y,u+v)=pr1(x,y,u+v)*x + (-pr2(x,y,u+v))*y by Def1 .= (pr1(x,y,u)+pr1(x,y,v))*x + (-pr2(x,y,u+v))*y by A1,Lm7 .=(pr1(x,y,u) + pr1(x,y,v))*x + (-(pr2(x,y,u) + pr2(x,y,v)))*y by A1,Lm7 .=pr1(x,y,u)*x + pr1(x,y,v)*x + (-(pr2(x,y,u) + pr2(x,y,v)))*y by RLVECT_1:def 9 .=pr1(x,y,u)*x + pr1(x,y,v)*x + (pr2(x,y,u) + pr2(x,y,v))*(-y) by RLVECT_1:38 .=pr1(x,y,u)*x + pr1(x,y,v)*x + (-((pr2(x,y,u) + pr2(x,y,v))*y)) by RLVECT_1:39 .=pr1(x,y,u)*x + pr1(x,y,v)*x + (-(pr2(x,y,u)*y +pr2(x,y,v)*y)) by RLVECT_1:def 9 .=pr1(x,y,u)*x + pr1(x,y,v)*x + (-(pr2(x,y,u)*y) + (-(pr2(x,y,v)*y))) by RLVECT_1:45 .=pr1(x,y,u)*x + (pr1(x,y,v)*x + (-(pr2(x,y,u)*y) + (-(pr2(x,y,v)*y)))) by RLVECT_1:def 6 .=pr1(x,y,u)*x + ((-(pr2(x,y,u)*y)) + (pr1(x,y,v)*x + (-pr2(x,y,v)*y))) by RLVECT_1:def 6 .=pr1(x,y,u)*x + (-(pr2(x,y,u)*y)) + (pr1(x,y,v)*x + (-pr2(x,y,v)*y)) by RLVECT_1:def 6 .=pr1(x,y,u)*x + (pr2(x,y,u)*(-y)) + (pr1(x,y,v)*x + (-pr2(x,y,v)*y)) by RLVECT_1:39 .=pr1(x,y,u)*x + (pr2(x,y,u)*(-y)) + (pr1(x,y,v)*x + (pr2(x,y,v)*(-y))) by RLVECT_1:39 .=pr1(x,y,u)*x + (-pr2(x,y,u))*y + (pr1(x,y,v)*x + (pr2(x,y,v)*(-y))) by RLVECT_1:38 .=pr1(x,y,u)*x + (-pr2(x,y,u))*y + (pr1(x,y,v)*x + (-pr2(x,y,v))*y) by RLVECT_1:38 .=pr1(x,y,u)*x + (-pr2(x,y,u))*y + Ortm(x,y,v) by Def1 .=Ortm(x,y,u) + Ortm(x,y,v) by Def1; end; theorem Th2: Gen x,y implies Ortm(x,y,n*u)= n*Ortm(x,y,u) proof assume A1:Gen x,y; thus Ortm(x,y,n*u)=pr1(x,y,n*u)*x + (-pr2(x,y,n*u))*y by Def1 .=n*pr1(x,y,u)*x + (-pr2(x,y,n*u))*y by A1,Lm7 .=n*pr1(x,y,u)*x + (-(n*pr2(x,y,u)))*y by A1,Lm7 .=n*pr1(x,y,u)*x + (n*pr2(x,y,u)*(-y)) by RLVECT_1:38 .=n*pr1(x,y,u)*x + (-(n*pr2(x,y,u)*y)) by RLVECT_1:39 .=n*pr1(x,y,u)*x + (-(n*(pr2(x,y,u)*y))) by RLVECT_1:def 9 .=n*pr1(x,y,u)*x + n*(-pr2(x,y,u)*y) by RLVECT_1:39 .=n*(pr1(x,y,u)*x) + n*(-pr2(x,y,u)*y) by RLVECT_1:def 9 .=n*((pr1(x,y,u)*x) + (-pr2(x,y,u)*y)) by RLVECT_1:def 9 .=n*((pr1(x,y,u)*x) + (pr2(x,y,u)*(-y))) by RLVECT_1:39 .=n*((pr1(x,y,u)*x) + ((-pr2(x,y,u))*y)) by RLVECT_1:38 .=n*Ortm(x,y,u) by Def1; end; theorem Gen x,y implies Ortm(x,y,0.V) = 0.V proof assume A1:Gen x,y; consider u; thus Ortm(x,y,0.V) = Ortm(x,y,0*u) by RLVECT_1:23 .= 0*Ortm(x,y,u) by A1,Th2 .= 0.V by RLVECT_1:23; end; theorem Th4: Gen x,y implies Ortm(x,y,-u) = -Ortm(x,y,u) proof assume A1:Gen x,y; then A2:-u=-(pr1(x,y,u)*x + pr2(x,y,u)*y) by Lm5 .=-(pr1(x,y,u)*x) + (-(pr2(x,y,u)*y)) by RLVECT_1:45 .=pr1(x,y,u)*(-x) + (-(pr2(x,y,u)*y)) by RLVECT_1:39 .=(-pr1(x,y,u))*x + (-(pr2(x,y,u)*y)) by RLVECT_1:38 .=(-pr1(x,y,u))*x + pr2(x,y,u)*(-y) by RLVECT_1:39 .=(-pr1(x,y,u))*x + (-pr2(x,y,u))*y by RLVECT_1:38; thus Ortm(x,y,-u)=pr1(x,y,-u)*x + (-pr2(x,y,-u))*y by Def1 .=(-pr1(x,y,u))*x + (-pr2(x,y,-u))*y by A1,A2,Lm6 .=(-pr1(x,y,u))*x + (-(-pr2(x,y,u)))*y by A1,A2,Lm6 .=pr1(x,y,u)*(-x) + (-(-pr2(x,y,u)))*y by RLVECT_1:38 .=-(pr1(x,y,u)*x) + (-(-pr2(x,y,u)))*y by RLVECT_1:39 .=-(pr1(x,y,u)*x) + (-pr2(x,y,u))*(-y) by RLVECT_1:38 .=-(pr1(x,y,u)*x) + (-((-pr2(x,y,u))*y)) by RLVECT_1:39 .=-(pr1(x,y,u)*x + (-pr2(x,y,u))*y) by RLVECT_1:45 .=-Ortm(x,y,u) by Def1; end; theorem Th5: Gen x,y implies Ortm(x,y,u-v)=Ortm(x,y,u)-Ortm(x,y,v) proof assume A1:Gen x,y; thus Ortm(x,y,u-v)=Ortm(x,y,u+(-v)) by RLVECT_1:def 11 .=Ortm(x,y,u) + Ortm(x,y,(-v)) by A1,Th1 .=Ortm(x,y,u) + (-Ortm(x,y,v)) by A1,Th4 .=Ortm(x,y,u) - Ortm(x,y,v) by RLVECT_1:def 11; end; theorem Th6: Gen x,y & Ortm(x,y,u)=Ortm(x,y,v) implies u=v proof assume A1:Gen x,y & Ortm(x,y,u)=Ortm(x,y,v); then Ortm(x,y,u)=pr1(x,y,v)*x + (-pr2(x,y,v))*y by Def1; then pr1(x,y,u)*x + (-pr2(x,y,u))*y - (pr1(x,y,v)*x + (-pr2(x,y,v))*y) =pr1(x,y,v)*x + (-pr2(x,y,v))*y - (pr1(x,y,v)*x + (-pr2(x,y,v))*y) by Def1; then pr1(x,y,u)*x + (-pr2(x,y,u))*y - (pr1(x,y,v)*x + (-pr2(x,y,v))*y) =0.V by RLVECT_1:28; then pr1(x,y,u)*x + (-pr2(x,y,u))*y - (pr1(x,y,v)*x) - (-pr2(x,y,v))*y =0.V by RLVECT_1:41; then pr1(x,y,u)*x + (-pr2(x,y,u))*y + (-(pr1(x,y,v)*x)) - (-pr2(x,y,v))*y =0.V by RLVECT_1:def 11; then pr1(x,y,u)*x + (-(pr1(x,y,v))*x) + ((-pr2(x,y,u))*y) - (-pr2(x,y,v))*y =0.V by RLVECT_1:def 6; then pr1(x,y,u)*x - pr1(x,y,v)*x + ((-pr2(x,y,u))*y) - (-pr2(x,y,v))*y =0.V by RLVECT_1:def 11; then pr1(x,y,u)*x - pr1(x,y,v)*x + ((-pr2(x,y,u))*y - (-pr2(x,y,v))*y) =0.V by RLVECT_1:42; then (pr1(x,y,u) - pr1(x,y,v))*x + ((-pr2(x,y,u))*y - (-pr2(x,y,v))*y) =0.V by RLVECT_1:49; then (pr1(x,y,u) - pr1(x,y,v))*x + ((-pr2(x,y,u)) - (-pr2(x,y,v)))*y =0.V by RLVECT_1:49; then pr1(x,y,u) - pr1(x,y,v)=0 & (-pr2(x,y,u)) - (-pr2(x,y,v))=0 by A1,ANALMETR:def 1; then A2: pr1(x,y,u)=pr1(x,y,v) & -pr2(x,y,u)=-pr2(x,y,v) by XCMPLX_1:15; hence u=pr1(x,y,v)*x + pr2(x,y,u)*y by A1,Lm5 .=pr1(x,y,v)*x + pr2(x,y,v)*y by A2,XCMPLX_1:134 .=v by A1,Lm5; end; theorem Th7: Gen x,y implies Ortm(x,y,Ortm(x,y,u))=u proof assume A1:Gen x,y; thus Ortm(x,y,Ortm(x,y,u))=Ortm(x,y,pr1(x,y,u)*x+(-pr2(x,y,u))*y) by Def1 .=pr1(x,y,pr1(x,y,u)*x+(-pr2(x,y,u))*y)*x +(-pr2(x,y,pr1(x,y,u)*x+(-pr2(x,y,u))*y))*y by Def1 .=pr1(x,y,u)*x+(-pr2(x,y,pr1(x,y,u)*x+(-pr2(x,y,u))*y))*y by A1,GEOMTRAP:def 4 .=pr1(x,y,u)*x+(-(-pr2(x,y,u)))*y by A1,GEOMTRAP:def 5 .=u by A1,Lm5; end; theorem Th8: Gen x,y implies ex v st u=Ortm(x,y,v) proof assume A1:Gen x,y; take Ortm(x,y,u); thus thesis by A1,Th7; end; definition let V,x,y; let u; func Orte(x,y,u) -> VECTOR of V equals :Def2: pr2(x,y,u)*x + (-pr1(x,y,u))*y; correctness; end; theorem Th9: Gen x,y implies Orte(x,y,-v)= -Orte(x,y,v) proof assume A1:Gen x,y; then A2:-v=-(pr1(x,y,v)*x + pr2(x,y,v)*y) by Lm5 .=-(pr1(x,y,v)*x) + (-(pr2(x,y,v)*y)) by RLVECT_1:45 .=pr1(x,y,v)*(-x) + (-(pr2(x,y,v)*y)) by RLVECT_1:39 .=(-pr1(x,y,v))*x + (-(pr2(x,y,v)*y)) by RLVECT_1:38 .=(-pr1(x,y,v))*x + pr2(x,y,v)*(-y) by RLVECT_1:39 .=(-pr1(x,y,v))*x + (-pr2(x,y,v))*y by RLVECT_1:38; thus Orte(x,y,-v)=pr2(x,y,-v)*x + (-pr1(x,y,-v))*y by Def2 .=(-pr2(x,y,v))*x + (-pr1(x,y,-v))*y by A1,A2,Lm6 .=(-pr2(x,y,v))*x + (-(-pr1(x,y,v)))*y by A1,A2,Lm6 .=pr2(x,y,v)*(-x) + (-(-pr1(x,y,v)))*y by RLVECT_1:38 .=-(pr2(x,y,v)*x) + (-(-pr1(x,y,v)))*y by RLVECT_1:39 .=-(pr2(x,y,v)*x) + (-pr1(x,y,v))*(-y) by RLVECT_1:38 .=-(pr2(x,y,v)*x) + (-((-pr1(x,y,v))*y)) by RLVECT_1:39 .=-(pr2(x,y,v)*x + (-pr1(x,y,v))*y) by RLVECT_1:45 .=-Orte(x,y,v) by Def2; end; theorem Th10: Gen x,y implies Orte(x,y,u+v)=Orte(x,y,u) + Orte(x,y,v) proof assume A1: Gen x,y; thus Orte(x,y,u+v)=pr2(x,y,u+v)*x + (-pr1(x,y,u+v))*y by Def2 .=(pr2(x,y,u+v))*x + (-(pr1(x,y,u)+pr1(x,y,v)))*y by A1,Lm7 .=(pr2(x,y,u)+pr2(x,y,v))*x + (-(pr1(x,y,u)+pr1(x,y,v)))*y by A1,Lm7 .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-(pr1(x,y,u)+pr1(x,y,v)))*y by RLVECT_1:def 9 .=pr2(x,y,u)*x + pr2(x,y,v)*x + (pr1(x,y,u)+pr1(x,y,v))*(-y) by RLVECT_1:38 .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-((pr1(x,y,u)+pr1(x,y,v))*y)) by RLVECT_1:39 .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-(pr1(x,y,u)*y +pr1(x,y,v)*y)) by RLVECT_1:def 9 .=pr2(x,y,u)*x + pr2(x,y,v)*x + (-(pr1(x,y,u)*y) +(-(pr1(x,y,v)*y))) by RLVECT_1:45 .=pr2(x,y,u)*x + (pr2(x,y,v)*x + (-(pr1(x,y,u)*y) +(-(pr1(x,y,v)*y)))) by RLVECT_1:def 6 .=pr2(x,y,u)*x + (-(pr1(x,y,u)*y) + (pr2(x,y,v)*x +(-(pr1(x,y,v)*y)))) by RLVECT_1:def 6 .=pr2(x,y,u)*x + (-(pr1(x,y,u)*y)) + (pr2(x,y,v)*x +(-(pr1(x,y,v)*y))) by RLVECT_1:def 6 .=pr2(x,y,u)*x + (pr1(x,y,u)*(-y)) + (pr2(x,y,v)*x +(-(pr1(x,y,v)*y))) by RLVECT_1:39 .=pr2(x,y,u)*x + (pr1(x,y,u)*(-y)) + (pr2(x,y,v)*x +(pr1(x,y,v)*(-y))) by RLVECT_1:39 .=pr2(x,y,u)*x + ((-pr1(x,y,u))*y) + (pr2(x,y,v)*x +(pr1(x,y,v)*(-y))) by RLVECT_1:38 .=pr2(x,y,u)*x + (-pr1(x,y,u))*y + (pr2(x,y,v)*x +(-pr1(x,y,v))*y) by RLVECT_1:38 .=pr2(x,y,u)*x + (-pr1(x,y,u))*y + Orte(x,y,v) by Def2 .=Orte(x,y,u) + Orte(x,y,v) by Def2; end; theorem Th11: Gen x,y implies Orte(x,y,u-v)=Orte(x,y,u)-Orte(x,y,v) proof assume A1:Gen x,y; thus Orte(x,y,u-v)=Orte(x,y,u+(-v)) by RLVECT_1:def 11 .=Orte(x,y,u) + Orte(x,y,(-v)) by A1,Th10 .=Orte(x,y,u) + (-Orte(x,y,v)) by A1,Th9 .=Orte(x,y,u) - Orte(x,y,v) by RLVECT_1:def 11; end; theorem Th12: Gen x,y implies Orte(x,y,n*u)=n*Orte(x,y,u) proof assume A1:Gen x,y; thus Orte(x,y,n*u)=pr2(x,y,n*u)*x + (-pr1(x,y,n*u))*y by Def2 .=n*pr2(x,y,u)*x + (-pr1(x,y,n*u))*y by A1,Lm7 .=n*pr2(x,y,u)*x + (-(n*pr1(x,y,u)))*y by A1,Lm7 .=n*pr2(x,y,u)*x + (n*pr1(x,y,u))*(-y) by RLVECT_1:38 .=n*pr2(x,y,u)*x + (-(n*pr1(x,y,u)*y)) by RLVECT_1:39 .=n*pr2(x,y,u)*x + (-(n*(pr1(x,y,u)*y))) by RLVECT_1:def 9 .=n*pr2(x,y,u)*x + n*(-(pr1(x,y,u)*y)) by RLVECT_1:39 .=n*(pr2(x,y,u)*x) + n*(-(pr1(x,y,u)*y)) by RLVECT_1:def 9 .=n*(pr2(x,y,u)*x + (-(pr1(x,y,u)*y))) by RLVECT_1:def 9 .=n*(pr2(x,y,u)*x + (pr1(x,y,u)*(-y))) by RLVECT_1:39 .=n*(pr2(x,y,u)*x + (-pr1(x,y,u))*y) by RLVECT_1:38 .=n*Orte(x,y,u) by Def2; end; theorem Th13: Gen x,y & Orte(x,y,u)=Orte(x,y,v) implies u=v proof assume A1:Gen x,y & Orte(x,y,u)=Orte(x,y,v); then Orte(x,y,u)=pr2(x,y,v)*x + (-pr1(x,y,v))*y by Def2; then pr2(x,y,u)*x + (-pr1(x,y,u))*y - (pr2(x,y,v)*x + (-pr1(x,y,v))*y) =pr2(x,y,v)*x + (-pr1(x,y,v))*y - (pr2(x,y,v)*x + (-pr1(x,y,v))*y) by Def2; then pr2(x,y,u)*x + (-pr1(x,y,u))*y - (pr2(x,y,v)*x + (-pr1(x,y,v))*y) =0.V by RLVECT_1:28; then pr2(x,y,u)*x + (-pr1(x,y,u))*y - (pr2(x,y,v)*x) - (-pr1(x,y,v))*y =0.V by RLVECT_1:41; then pr2(x,y,u)*x + (-pr1(x,y,u))*y + (-(pr2(x,y,v)*x)) - (-pr1(x,y,v))*y =0.V by RLVECT_1:def 11; then pr2(x,y,u)*x + (-(pr2(x,y,v))*x) + ((-pr1(x,y,u))*y) - (-pr1(x,y,v))*y =0.V by RLVECT_1:def 6; then pr2(x,y,u)*x - pr2(x,y,v)*x + ((-pr1(x,y,u))*y) - (-pr1(x,y,v))*y =0.V by RLVECT_1:def 11; then pr2(x,y,u)*x - pr2(x,y,v)*x + ((-pr1(x,y,u))*y - (-pr1(x,y,v))*y) =0.V by RLVECT_1:42; then (pr2(x,y,u) - pr2(x,y,v))*x + ((-pr1(x,y,u))*y - (-pr1(x,y,v))*y) =0.V by RLVECT_1:49; then (pr2(x,y,u) - pr2(x,y,v))*x + ((-pr1(x,y,u)) - (-pr1(x,y,v)))*y =0.V by RLVECT_1:49; then pr2(x,y,u) - pr2(x,y,v)=0 & (-pr1(x,y,u)) - (-pr1(x,y,v))=0 by A1,ANALMETR:def 1; then A2: pr2(x,y,u)=pr2(x,y,v) & -pr1(x,y,u)=-pr1(x,y,v) by XCMPLX_1:15; thus u=pr1(x,y,u)*x + pr2(x,y,u)*y by A1,Lm5 .=pr1(x,y,v)*x + pr2(x,y,v)*y by A2,XCMPLX_1:134 .=v by A1,Lm5; end; theorem Th14: Gen x,y implies Orte(x,y,Orte(x,y,u)) = -u proof assume A1:Gen x,y; thus Orte(x,y,Orte(x,y,u))=Orte(x,y,pr2(x,y,u)*x+(-pr1(x,y,u))*y) by Def2 .=(pr2(x,y,pr2(x,y,u)*x+(-pr1(x,y,u))*y))*x+ (-pr1(x,y,pr2(x,y,u)*x+(-pr1(x,y,u))*y))*y by Def2 .=(-pr1(x,y,u))*x+ (-pr1(x,y,pr2(x,y,u)*x+(-pr1(x,y,u))*y))*y by A1,GEOMTRAP:def 5 .=(-pr1(x,y,u))*x+(-pr2(x,y,u))*y by A1,GEOMTRAP:def 4 .=pr1(x,y,u)*(-x)+(-pr2(x,y,u))*y by RLVECT_1:38 .=-(pr1(x,y,u)*x)+(-pr2(x,y,u))*y by RLVECT_1:39 .=-(pr1(x,y,u)*x)+pr2(x,y,u)*(-y) by RLVECT_1:38 .=-(pr1(x,y,u)*x)+(-(pr2(x,y,u)*y)) by RLVECT_1:39 .=-(pr1(x,y,u)*x+pr2(x,y,u)*y) by RLVECT_1:45 .=-u by A1,Lm5; end; theorem Th15: Gen x,y implies ex v st Orte(x,y,v) = u proof assume A1:Gen x,y; take v= -Orte(x,y,u); thus Orte(x,y,v) = -Orte(x,y,Orte(x,y,u)) by A1,Th9 .= -(-u) by A1,Th14 .= u by RLVECT_1:30; end; definition let V; let x,y,u,v,u1,v1; pred u,v,u1,v1 are_COrte_wrt x,y means :Def3: Orte(x,y,u),Orte(x,y,v) // u1,v1; pred u,v,u1,v1 are_COrtm_wrt x,y means :Def4: Ortm(x,y,u),Ortm(x,y,v) // u1,v1; end; theorem Th16: Gen x,y implies (u,v // u1,v1 implies Orte(x,y,u),Orte(x,y,v) // Orte(x,y,u1),Orte(x,y,v1)) proof assume A1: Gen x,y; assume A2: u,v // u1,v1; now assume u<>v & u1<>v1; then consider a,b such that A3: 0<a & 0<b and A4:a*(v-u)=b*(v1-u1) by A2,ANALOAF:def 1; a*(Orte(x,y,v)-Orte(x,y,u)) = a*(Orte(x,y,v-u)) by A1,Th11 .= Orte(x,y,b*(v1-u1)) by A1,A4,Th12 .= b*(Orte(x,y,v1-u1)) by A1,Th12 .=b*(Orte(x,y,v1)-Orte(x,y,u1)) by A1,Th11; hence Orte(x,y,u),Orte(x,y,v) // Orte(x,y,u1),Orte(x,y,v1) by A3,ANALOAF:def 1;end; hence thesis by ANALOAF:18;end; theorem Th17: Gen x,y implies (u,v // u1,v1 implies Ortm(x,y,u),Ortm(x,y,v) // Ortm(x,y,u1),Ortm(x,y,v1)) proof assume A1: Gen x,y; assume A2: u,v // u1,v1; now assume A3:u<>v; now assume u1<>v1; then consider a,b such that A4: 0<a & 0<b and A5:a*(v-u)=b*(v1-u1) by A2,A3,ANALOAF:def 1; a*(Ortm(x,y,v)-Ortm(x,y,u)) = a*(Ortm(x,y,v-u)) by A1,Th5 .= Ortm(x,y,b*(v1-u1)) by A1,A5,Th2 .= b*(Ortm(x,y,v1-u1)) by A1,Th2 .=b*(Ortm(x,y,v1)-Ortm(x,y,u1)) by A1,Th5; hence thesis by A4,ANALOAF:def 1;end; hence thesis by ANALOAF:18;end; hence thesis by ANALOAF:18;end; theorem Th18: Gen x,y implies (u,u1,v,v1 are_COrte_wrt x,y implies v,v1,u1,u are_COrte_wrt x,y) proof assume A1:Gen x,y;assume u,u1,v,v1 are_COrte_wrt x,y; then Orte(x,y,u),Orte(x,y,u1) // v,v1 by Def3; then v,v1 // Orte(x,y,u),Orte(x,y,u1) by ANALOAF:21; then Orte(x,y,v),Orte(x,y,v1) // Orte(x,y,Orte(x,y,u)),Orte(x,y,Orte(x,y,u1)) by A1,Th16; then Orte(x,y,v),Orte(x,y,v1) // -u,Orte(x,y,Orte(x,y,u1)) by A1,Th14; then Orte(x,y,v),Orte(x,y,v1) // -u,-u1 by A1,Th14; then A2:-u,-u1 // Orte(x,y,v),Orte(x,y,v1) by ANALOAF:21; -u1-(-u)=-u1+(-(-u)) by RLVECT_1:def 11 .=u+(-u1) by RLVECT_1:30 .=u-u1 by RLVECT_1:def 11; then A3:-u,-u1 // u1,u by ANALOAF:24; A4:now assume -u<>-u1; then Orte(x,y,v),Orte(x,y,v1) // u1,u by A2,A3,ANALOAF:20; hence thesis by Def3;end; now assume -u=-u1; then u=-(-u1) by RLVECT_1:30 .= u1 by RLVECT_1:30; then Orte(x,y,v),Orte(x,y,v1) // u1,u by ANALOAF:18; hence thesis by Def3;end; hence thesis by A4;end; theorem Th19: Gen x,y implies (u,u1,v,v1 are_COrtm_wrt x,y implies v,v1,u,u1 are_COrtm_wrt x,y) proof assume A1:Gen x,y;assume u,u1,v,v1 are_COrtm_wrt x,y; then Ortm(x,y,u),Ortm(x,y,u1) // v,v1 by Def4; then v,v1 // Ortm(x,y,u),Ortm(x,y,u1) by ANALOAF:21; then Ortm(x,y,v),Ortm(x,y,v1) // Ortm(x,y,Ortm(x,y,u)),Ortm(x,y,Ortm(x,y,u1)) by A1,Th17; then Ortm(x,y,v),Ortm(x,y,v1) // u,Ortm(x,y,Ortm(x,y,u1)) by A1,Th7; then Ortm(x,y,v),Ortm(x,y,v1) // u,u1 by A1,Th7; hence thesis by Def4;end; theorem Th20: u,u,v,w are_COrte_wrt x,y proof Orte(x,y,u),Orte(x,y,u) // v,w by ANALOAF:18; hence u,u,v,w are_COrte_wrt x,y by Def3; end; theorem u,u,v,w are_COrtm_wrt x,y proof Ortm(x,y,u),Ortm(x,y,u) // v,w by ANALOAF:18; hence u,u,v,w are_COrtm_wrt x,y by Def4; end; theorem u,v,w,w are_COrte_wrt x,y proof Orte(x,y,u),Orte(x,y,v) // w,w by ANALOAF:18; hence u,v,w,w are_COrte_wrt x,y by Def3; end; theorem u,v,w,w are_COrtm_wrt x,y proof Ortm(x,y,u),Ortm(x,y,v) // w,w by ANALOAF:18; hence u,v,w,w are_COrtm_wrt x,y by Def4; end; theorem Th24: Gen x,y implies u,v,Orte(x,y,u),Orte(x,y,v) are_Ort_wrt x,y proof assume A1:Gen x,y; set w = Orte(x,y,v) - Orte(x,y,u); A2:w = Orte(x,y,v-u) by A1,Th11 .= pr2(x,y,v-u)*x + (-pr1(x,y,v-u))*y by Def2; PProJ(x,y,v-u,w) = pr1(x,y,v-u)*pr1(x,y,w) + pr2(x,y,v-u)*pr2(x,y,w) by GEOMTRAP:def 6 .= pr1(x,y,v-u)*pr2(x,y,v-u) + pr2(x,y,v-u)*pr2(x,y,w) by A1,A2,Lm6 .= pr1(x,y,v-u)*pr2(x,y,v-u) + (-pr1(x,y,v-u))*pr2(x,y,v-u) by A1,A2,Lm6 .= pr1(x,y,v-u)*pr2(x,y,v-u) + (-pr1(x,y,v-u)*pr2(x,y,v-u)) by XCMPLX_1:175 .= 0 by XCMPLX_0:def 6; then v-u,w are_Ort_wrt x,y by A1,GEOMTRAP:34; hence thesis by ANALMETR:def 3; end; theorem u,v,Orte(x,y,u),Orte(x,y,v) are_COrte_wrt x,y proof Orte(x,y,u),Orte(x,y,v) // Orte(x,y,u),Orte(x,y,v) by ANALOAF:17; hence u,v,Orte(x,y,u),Orte(x,y,v) are_COrte_wrt x,y by Def3; end; theorem u,v,Ortm(x,y,u),Ortm(x,y,v) are_COrtm_wrt x,y proof Ortm(x,y,u),Ortm(x,y,v) // Ortm(x,y,u),Ortm(x,y,v) by ANALOAF:17; hence u,v,Ortm(x,y,u),Ortm(x,y,v) are_COrtm_wrt x,y by Def4; end; theorem Gen x,y implies (u,v // u1,v1 iff ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y) proof assume A1:Gen x,y; (u,v // u1,v1 iff ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y) proof A2:u,v // u1,v1 implies ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y proof assume A3:u,v // u1,v1; A4:now assume A5: u=v & u1=v1; take u2=0.V,v2=x; Orte(x,y,u2),Orte(x,y,v2) // u,v & Orte(x,y,u2),Orte(x,y,v2) // u1,v1 by A5,ANALOAF:18; then A6:u2,v2,u1,v1 are_COrte_wrt x,y & u2,v2,u,v are_COrte_wrt x,y by Def3; u2<>v2 by A1,Lm4; hence thesis by A6; end; A7:now assume A8: u<>v; consider u2 such that A9: Orte(x,y,u2)=u by A1,Th15; consider v2 such that A10: Orte(x,y,v2)=v by A1,Th15; Orte(x,y,u2),Orte(x,y,v2) // u,v by A9,A10,ANALOAF:17; then A11:u2,v2,u,v are_COrte_wrt x,y by Def3; u2,v2,u1,v1 are_COrte_wrt x,y by A3,A9,A10,Def3; hence thesis by A8,A9,A10,A11; end; now assume A12: u1<>v1; consider u2 such that A13: Orte(x,y,u2)=u1 by A1,Th15; consider v2 such that A14: Orte(x,y,v2)=v1 by A1,Th15; Orte(x,y,u2),Orte(x,y,v2) // u1,v1 by A13,A14,ANALOAF:17; then A15:u2,v2,u1,v1 are_COrte_wrt x,y by Def3; Orte(x,y,u2),Orte(x,y,v2) // u,v by A3,A13,A14,ANALOAF:21; then u2,v2,u,v are_COrte_wrt x,y by Def3; hence thesis by A12,A13,A14,A15; end; hence thesis by A4,A7; end; (ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y) implies u,v // u1,v1 proof given u2,v2 such that A16:u2<>v2 & u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y; A17: Orte(x,y,u2) <> Orte(x,y,v2) by A1,A16,Th13; Orte(x,y,u2),Orte(x,y,v2) // u,v & Orte(x,y,u2),Orte(x,y,v2) // u1,v1 by A16,Def3; hence thesis by A17,ANALOAF:20; end; hence thesis by A2; end; hence thesis; end; theorem Gen x,y implies (u,v // u1,v1 iff ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrtm_wrt x,y & u2,v2,u1,v1 are_COrtm_wrt x,y) proof assume A1:Gen x,y; (u,v // u1,v1 iff ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrtm_wrt x,y & u2,v2,u1,v1 are_COrtm_wrt x,y) proof A2:u,v // u1,v1 implies ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrtm_wrt x,y & u2,v2,u1,v1 are_COrtm_wrt x,y proof assume A3:u,v // u1,v1; A4:now assume A5: u=v & u1=v1; take u2=0.V,v2=x; Ortm(x,y,u2),Ortm(x,y,v2) // u,v & Ortm(x,y,u2),Ortm(x,y,v2) // u1,v1 by A5,ANALOAF:18; then A6:u2,v2,u1,v1 are_COrtm_wrt x,y & u2,v2,u,v are_COrtm_wrt x,y by Def4; u2<>v2 by A1,Lm4; hence thesis by A6; end; A7:now assume A8: u<>v; consider u2 such that A9: Ortm(x,y,u2)=u by A1,Th8; consider v2 such that A10: Ortm(x,y,v2)=v by A1,Th8; Ortm(x,y,u2),Ortm(x,y,v2) // u,v by A9,A10,ANALOAF:17; then A11:u2,v2,u,v are_COrtm_wrt x,y by Def4; u2,v2,u1,v1 are_COrtm_wrt x,y by A3,A9,A10,Def4; hence thesis by A8,A9,A10,A11; end; now assume A12: u1<>v1; consider u2 such that A13: Ortm(x,y,u2)=u1 by A1,Th8; consider v2 such that A14: Ortm(x,y,v2)=v1 by A1,Th8; Ortm(x,y,u2),Ortm(x,y,v2) // u1,v1 by A13,A14,ANALOAF:17; then A15:u2,v2,u1,v1 are_COrtm_wrt x,y by Def4; Ortm(x,y,u2),Ortm(x,y,v2) // u,v by A3,A13,A14,ANALOAF:21; then u2,v2,u,v are_COrtm_wrt x,y by Def4; hence thesis by A12,A13,A14,A15; end; hence thesis by A4,A7; end; (ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrtm_wrt x,y & u2,v2,u1,v1 are_COrtm_wrt x,y) implies u,v // u1,v1 proof given u2,v2 such that A16:u2<>v2 & u2,v2,u,v are_COrtm_wrt x,y & u2,v2,u1,v1 are_COrtm_wrt x,y; A17: Ortm(x,y,u2) <> Ortm(x,y,v2) by A1,A16,Th6; Ortm(x,y,u2),Ortm(x,y,v2) // u,v & Ortm(x,y,u2),Ortm(x,y,v2) // u1,v1 by A16,Def4; hence thesis by A17,ANALOAF:20; end; hence thesis by A2; end; hence thesis; end; theorem Gen x,y implies (u,v,u1,v1 are_Ort_wrt x,y iff u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y) proof assume A1:Gen x,y; A2: now assume u=v; then v-u=0.V by RLVECT_1:28; then v-u,v1-u1 are_Ort_wrt x,y by A1,ANALMETR:9; hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3; end; now assume A3: u<>v; set u2=Orte(x,y,u),v2=Orte(x,y,v); A4: v-u<>0.V by A3,RLVECT_1:35; u,v,u2,v2 are_Ort_wrt x,y by A1,Th24; then A5: v-u,v2-u2 are_Ort_wrt x,y by ANALMETR:def 3; A6: now assume u,v,u1,v1 are_Ort_wrt x,y; then v-u,v1-u1 are_Ort_wrt x,y by ANALMETR:def 3; then ex a,b st a*(v2-u2)=b*(v1-u1) & (a<>0 or b<>0) by A1,A4,A5,ANALMETR:13; then u2,v2 // u1,v1 or u2,v2 // v1,u1 by ANALMETR:18; hence u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y by Def3; end; now assume u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y; then u2,v2 // u1,v1 or u2,v2 // v1,u1 by Def3; then consider a,b such that A7: a*(v2-u2)=b*(v1-u1) and A8: a<>0 or b<>0 by ANALMETR:18; A9: now assume A10: b=0; then 0.V = a*(v2-u2) by A7,RLVECT_1:23; then v2-u2=0.V by A8,A10,RLVECT_1:24; then v2=u2 by RLVECT_1:35; then u=v by A1,Th13; then v-u=0.V by RLVECT_1:28; then v-u,v1-u1 are_Ort_wrt x,y by A1,ANALMETR:9; hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3; end; now assume A11: b<>0; ((b")*a)*(v2-u2)=(b")*(b*(v1-u1)) by A7,RLVECT_1:def 9; then ((b")*a)*(v2-u2)=((b")*b)*(v1-u1) by RLVECT_1:def 9; then ((b")*a)*(v2-u2)=1*(v1-u1) by A11,XCMPLX_0:def 7; then v1-u1=((b")*a)*(v2-u2) by RLVECT_1:def 9; then v-u,v1-u1 are_Ort_wrt x,y by A5,ANALMETR:11; hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3; end; hence u,v,u1,v1 are_Ort_wrt x,y by A9; end; hence thesis by A6; end; hence thesis by A2,Th20; end; theorem Gen x,y & u,v,u1,v1 are_COrte_wrt x,y & u,v,v1,u1 are_COrte_wrt x,y implies u=v or u1=v1 proof assume A1:Gen x,y; assume A2:u,v,u1,v1 are_COrte_wrt x,y & u,v,v1,u1 are_COrte_wrt x,y; assume A3: u<>v & u1<>v1; A4: Orte(x,y,u),Orte(x,y,v) // u1,v1 & Orte(x,y,u),Orte(x,y,v) // v1,u1 by A2,Def3; Orte(x,y,u) <> Orte(x,y,v) by A1,A3,Th13; then u1,v1 // v1,u1 by A4,ANALOAF:20; hence contradiction by A3,ANALOAF:19; end; theorem Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,u1 are_COrtm_wrt x,y implies u=v or u1=v1 proof assume A1:Gen x,y; assume A2:u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,u1 are_COrtm_wrt x,y; assume A3: u<>v & u1<>v1; A4: Ortm(x,y,u),Ortm(x,y,v) // u1,v1 & Ortm(x,y,u),Ortm(x,y,v) // v1,u1 by A2,Def4; Ortm(x,y,u) <> Ortm(x,y,v) by A1,A3,Th6; then u1,v1 // v1,u1 by A4,ANALOAF:20; hence contradiction by A3,ANALOAF:19; end; theorem Gen x,y & u,v,u1,v1 are_COrte_wrt x,y & u,v,u1,w are_COrte_wrt x,y implies u,v,v1,w are_COrte_wrt x,y or u,v,w,v1 are_COrte_wrt x,y proof assume A1:Gen x,y & u,v,u1,v1 are_COrte_wrt x,y & u,v,u1,w are_COrte_wrt x,y; then A2:Orte(x,y,u),Orte(x,y,v) // u1,v1 & Orte(x,y,u),Orte(x,y,v) // u1,w by Def3; now assume A3:u<>v; now assume A4: u1<>v1 & u1<>w; Orte(x,y,u) <> Orte(x,y,v) by A1,A3,Th13; then A5: u1,v1 // u1,w by A2,ANALOAF:20; A6: now assume A7: u1,v1 // v1,w; u1,v1 // Orte(x,y,u),Orte(x,y,v) by A2,ANALOAF:21; then Orte(x,y,u),Orte(x,y,v) // v1,w by A4,A7,ANALOAF:20; hence thesis by Def3; end; now assume A8: u1,w // w,v1; u1,w // Orte(x,y,u),Orte(x,y,v) by A2,ANALOAF:21; then Orte(x,y,u),Orte(x,y,v) // w,v1 by A4,A8,ANALOAF:20; hence thesis by Def3; end; hence thesis by A5,A6,ANALOAF:23; end; hence thesis by A1; end; hence thesis by Th20; end; theorem Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,u1,w are_COrtm_wrt x,y implies u,v,v1,w are_COrtm_wrt x,y or u,v,w,v1 are_COrtm_wrt x,y proof assume A1:Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,u1,w are_COrtm_wrt x,y; then A2:Ortm(x,y,u),Ortm(x,y,v) // u1,v1 & Ortm(x,y,u),Ortm(x,y,v) // u1,w by Def4; A3:now assume A4:u<>v & u1<>v1; then Ortm(x,y,u) <> Ortm(x,y,v) by A1,Th6; then u1,v1 // u1,w by A2,ANALOAF:20; then A5:u1,v1 // v1,w or u1,w // w,v1 by ANALOAF:23; now assume A6:u1<>w; u1,v1 // Ortm(x,y,u),Ortm(x,y,v) & u1,w // Ortm(x,y,u),Ortm(x,y,v) by A2,ANALOAF:21; then Ortm(x,y,u),Ortm(x,y,v) // v1,w or Ortm(x,y,u),Ortm(x,y,v) // w,v1 by A4,A5,A6,ANALOAF:20; hence thesis by Def4;end; hence thesis by A1;end; now assume u=v; then Ortm(x,y,u),Ortm(x,y,v) // v1,w or Ortm(x,y,u),Ortm(x,y,v) // w,v1 by ANALOAF:18; hence thesis by Def4;end; hence thesis by A1,A3;end; theorem u,v,u1,v1 are_COrte_wrt x,y implies v,u,v1,u1 are_COrte_wrt x,y proof assume u,v,u1,v1 are_COrte_wrt x,y; then Orte(x,y,u),Orte(x,y,v) // u1,v1 by Def3; then Orte(x,y,v),Orte(x,y,u) // v1,u1 by ANALOAF:21; hence thesis by Def3; end; theorem u,v,u1,v1 are_COrtm_wrt x,y implies v,u,v1,u1 are_COrtm_wrt x,y proof assume u,v,u1,v1 are_COrtm_wrt x,y; then Ortm(x,y,u),Ortm(x,y,v) // u1,v1 by Def4; then Ortm(x,y,v),Ortm(x,y,u) // v1,u1 by ANALOAF:21; hence thesis by Def4; end; theorem Gen x,y & u,v,u1,v1 are_COrte_wrt x,y & u,v,v1,w are_COrte_wrt x,y implies u,v,u1,w are_COrte_wrt x,y proof assume A1:Gen x,y; assume A2:u,v,u1,v1 are_COrte_wrt x,y & u,v,v1,w are_COrte_wrt x,y; then A3:Orte(x,y,u),Orte(x,y,v) // u1,v1 & Orte(x,y,u),Orte(x,y,v) // v1,w by Def3; then A4:u1,v1 // Orte(x,y,u),Orte(x,y,v) by ANALOAF:21; A5:now assume u<>v; then Orte(x,y,u) <> Orte(x,y,v) by A1,Th13; then u1,v1 // v1,w by A3,ANALOAF:20; then A6:u1,v1 // u1,w by ANALOAF:22; now assume u1<>v1; then Orte(x,y,u),Orte(x,y,v) // u1,w by A4,A6,ANALOAF:20; hence thesis by Def3;end; hence thesis by A2; end; now assume u=v; then Orte(x,y,u),Orte(x,y,v) // u1,w by ANALOAF:18; hence thesis by Def3; end; hence thesis by A5; end; theorem Gen x,y & u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,w are_COrtm_wrt x,y implies u,v,u1,w are_COrtm_wrt x,y proof assume A1:Gen x,y; assume A2:u,v,u1,v1 are_COrtm_wrt x,y & u,v,v1,w are_COrtm_wrt x,y; then A3:Ortm(x,y,u),Ortm(x,y,v) // u1,v1 & Ortm(x,y,u),Ortm(x,y,v) // v1,w by Def4; then A4:u1,v1 // Ortm(x,y,u),Ortm(x,y,v) by ANALOAF:21; A5:now assume u<>v; then Ortm(x,y,u) <> Ortm(x,y,v) by A1,Th6; then u1,v1 // v1,w by A3,ANALOAF:20; then A6: u1,v1 // u1,w by ANALOAF:22; now assume u1<>v1; then Ortm(x,y,u),Ortm(x,y,v) // u1,w by A4,A6,ANALOAF:20; hence thesis by Def4; end; hence thesis by A2; end; now assume u=v; then Ortm(x,y,u),Ortm(x,y,v) // u1,w by ANALOAF:18; hence thesis by Def4; end; hence thesis by A5; end; theorem Gen x,y implies for u,v,w ex u1 st w<>u1 & w,u1,u,v are_COrte_wrt x,y proof assume A1:Gen x,y;let u,v,w; A2:now assume A3: u=v; take u1=w+x; Orte(x,y,w),Orte(x,y,u1) // u,v by A3,ANALOAF:18; then A4:w,u1,u,v are_COrte_wrt x,y by Def3; now assume w=u1; then x=0.V by RLVECT_1:22; hence contradiction by A1,Lm4; end; hence thesis by A4; end; now assume A5:u<>v; consider u2 such that A6: Orte(x,y,u2)=u by A1,Th15; consider v2 such that A7: Orte(x,y,v2)=v by A1,Th15; take u1= (v2+w)-u2; u2,v2 // w,u1 by ANALOAF:25; then w,u1 // u2,v2 by ANALOAF:21; then Orte(x,y,w),Orte(x,y,u1) // Orte(x,y,u2),Orte(x,y,v2) by A1,Th16; then A8:w,u1,u,v are_COrte_wrt x,y by A6,A7,Def3; now assume w=u1; then w= w+(v2-u2) by RLVECT_1:42; then v2-u2=0.V by RLVECT_1:22; hence contradiction by A5,A6,A7,RLVECT_1:35; end; hence thesis by A8; end; hence thesis by A2; end; theorem Gen x,y implies for u,v,w ex u1 st w<>u1 & w,u1,u,v are_COrtm_wrt x,y proof assume A1:Gen x,y; let u,v,w; A2: now assume A3:u=v; take u1=w+x; Ortm(x,y,w),Ortm(x,y,u1) // u,v by A3,ANALOAF:18; then A4:w,u1,u,v are_COrtm_wrt x,y by Def4; now assume w=u1; then x=0.V by RLVECT_1:22; hence contradiction by A1,Lm4; end; hence thesis by A4; end; now assume A5:u<>v; consider u2 such that A6: Ortm(x,y,u2)=u by A1,Th8; consider v2 such that A7: Ortm(x,y,v2)=v by A1,Th8; take u1= (v2+w)-u2; u2,v2 // w,u1 by ANALOAF:25; then w,u1 // u2,v2 by ANALOAF:21; then Ortm(x,y,w),Ortm(x,y,u1) // Ortm(x,y,u2),Ortm(x,y,v2) by A1,Th17; then A8:w,u1,u,v are_COrtm_wrt x,y by A6,A7,Def4; now assume w=u1; then w= w+(v2-u2) by RLVECT_1:42; then v2-u2=0.V by RLVECT_1:22; hence contradiction by A5,A6,A7,RLVECT_1:35; end; hence thesis by A8; end; hence thesis by A2; end; theorem Th40: Gen x,y implies for u,v,w ex u1 st w<>u1 & u,v,w,u1 are_COrte_wrt x,y proof assume A1:Gen x,y;let u,v,w; A2:now assume A3: u=v; take u1=w+x; Orte(x,y,u),Orte(x,y,v) // w,u1 by A3,ANALOAF:18; then A4:u,v,w,u1 are_COrte_wrt x,y by Def3; now assume w=u1; then x=0.V by RLVECT_1:22; hence contradiction by A1,Lm4; end; hence thesis by A4; end; now assume A5:u<>v; consider u2 such that A6: Orte(x,y,u2)=u by A1,Th15; consider v2 such that A7: Orte(x,y,v2)=v by A1,Th15; take u1= (u2+w)-v2; v2,u2 // w,u1 by ANALOAF:25; then Orte(x,y,v2),Orte(x,y,u2) // Orte(x,y,w),Orte(x,y,u1) by A1,Th16; then Orte(x,y,w),Orte(x,y,u1) // v,u by A6,A7,ANALOAF:21; then Orte(x,y,u1),Orte(x,y,w) // u,v by ANALOAF:21; then u1,w,u,v are_COrte_wrt x,y by Def3; then A8:u,v,w,u1 are_COrte_wrt x,y by A1,Th18; now assume w=u1; then w= w+(u2-v2) by RLVECT_1:42; then u2-v2=0.V by RLVECT_1:22; hence contradiction by A5,A6,A7,RLVECT_1:35; end; hence thesis by A8; end; hence thesis by A2; end; theorem Th41: Gen x,y implies for u,v,w ex u1 st w<>u1 & u,v,w,u1 are_COrtm_wrt x,y proof assume A1:Gen x,y; let u,v,w; A2: now assume A3:u=v; take u1=w+x; Ortm(x,y,w),Ortm(x,y,u1) // u,v by A3,ANALOAF:18; then w,u1,u,v are_COrtm_wrt x,y by Def4; then A4:u,v,w,u1 are_COrtm_wrt x,y by A1,Th19; w<>u1 proof assume w=u1; then x=0.V by RLVECT_1:22; hence contradiction by A1,Lm4; end; hence thesis by A4; end; now assume A5:u<>v; consider u2 such that A6: Ortm(x,y,u2)=u by A1,Th8; consider v2 such that A7: Ortm(x,y,v2)=v by A1,Th8; take u1= (v2+w)-u2; u2,v2 // w,u1 by ANALOAF:25; then w,u1 // u2,v2 by ANALOAF:21; then Ortm(x,y,w),Ortm(x,y,u1) // Ortm(x,y,u2),Ortm(x,y,v2) by A1,Th17; then w,u1,u,v are_COrtm_wrt x,y by A6,A7,Def4; then A8:u,v,w,u1 are_COrtm_wrt x,y by A1,Th19; w<>u1 proof assume w=u1; then w= w+(v2-u2) by RLVECT_1:42; then v2-u2=0.V by RLVECT_1:22; hence contradiction by A5,A6,A7,RLVECT_1:35; end; hence thesis by A8;end; hence thesis by A2; end; theorem Gen x,y & u,u1,v,v1 are_COrte_wrt x,y & w,w1,v,v1 are_COrte_wrt x,y & w,w1,u2,v2 are_COrte_wrt x,y implies w=w1 or v=v1 or u,u1,u2,v2 are_COrte_wrt x,y proof assume A1:Gen x,y;assume A2:u,u1,v,v1 are_COrte_wrt x,y & w,w1,v,v1 are_COrte_wrt x,y & w,w1,u2,v2 are_COrte_wrt x,y; then Orte(x,y,u),Orte(x,y,u1) // v,v1 by Def3; then A3:v,v1 // Orte(x,y,u),Orte(x,y,u1) by ANALOAF:21; Orte(x,y,w),Orte(x,y,w1) // v,v1 by A2,Def3; then A4:v,v1 // Orte(x,y,w),Orte(x,y,w1) by ANALOAF:21; A5:Orte(x,y,w),Orte(x,y,w1) // u2,v2 by A2,Def3; now assume A6:w<>w1 & v<>v1; then A7:Orte(x,y,w) <> Orte(x,y,w1) by A1,Th13; Orte(x,y,w),Orte(x,y,w1) // Orte(x,y,u),Orte(x,y,u1) by A3,A4,A6,ANALOAF:20; then Orte(x,y,u),Orte(x,y,u1) // u2,v2 by A5,A7,ANALOAF:20; hence u,u1,u2,v2 are_COrte_wrt x,y by Def3;end; hence thesis;end; theorem Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & w,w1,v,v1 are_COrtm_wrt x,y & w,w1,u2,v2 are_COrtm_wrt x,y implies w=w1 or v=v1 or u,u1,u2,v2 are_COrtm_wrt x,y proof assume A1:Gen x,y;assume A2:u,u1,v,v1 are_COrtm_wrt x,y & w,w1,v,v1 are_COrtm_wrt x,y & w,w1,u2,v2 are_COrtm_wrt x,y; then Ortm(x,y,u),Ortm(x,y,u1) // v,v1 by Def4; then A3:v,v1 // Ortm(x,y,u),Ortm(x,y,u1) by ANALOAF:21; Ortm(x,y,w),Ortm(x,y,w1) // v,v1 by A2,Def4; then A4:v,v1 // Ortm(x,y,w),Ortm(x,y,w1) by ANALOAF:21; A5:Ortm(x,y,w),Ortm(x,y,w1) // u2,v2 by A2,Def4; now assume A6:w<>w1 & v<>v1; then A7:Ortm(x,y,w) <> Ortm(x,y,w1) by A1,Th6; Ortm(x,y,w),Ortm(x,y,w1) // Ortm(x,y,u),Ortm(x,y,u1) by A3,A4,A6,ANALOAF:20; then Ortm(x,y,u),Ortm(x,y,u1) // u2,v2 by A5,A7,ANALOAF:20; hence u,u1,u2,v2 are_COrtm_wrt x,y by Def4;end; hence thesis;end; canceled 2; theorem Gen x,y & u,u1,v,v1 are_COrte_wrt x,y & v,v1,w,w1 are_COrte_wrt x,y & u2,v2,w,w1 are_COrte_wrt x,y implies u,u1,u2,v2 are_COrte_wrt x,y or v=v1 or w=w1 proof assume A1:Gen x,y;assume A2:u,u1,v,v1 are_COrte_wrt x,y & v,v1,w,w1 are_COrte_wrt x,y & u2,v2,w,w1 are_COrte_wrt x,y; then v,v1,u1,u are_COrte_wrt x,y by A1,Th18; then A3:Orte(x,y,v),Orte(x,y,v1) // u1,u by Def3; Orte(x,y,v),Orte(x,y,v1) // w,w1 by A2,Def3; then A4:w,w1 // Orte(x,y,v),Orte(x,y,v1) by ANALOAF:21; Orte(x,y,u2),Orte(x,y,v2) // w,w1 by A2,Def3; then A5:w,w1 // Orte(x,y,u2),Orte(x,y,v2) by ANALOAF:21; now assume A6:w<>w1 & v<>v1; then A7:Orte(x,y,v),Orte(x,y,v1) // Orte(x,y,u2),Orte(x,y,v2) by A4,A5,ANALOAF:20; Orte(x,y,v) <> Orte(x,y,v1) by A1,A6,Th13; then Orte(x,y,u2),Orte(x,y,v2) // u1,u by A3,A7,ANALOAF:20; then Orte(x,y,v2),Orte(x,y,u2) // u,u1 by ANALOAF:21; then v2,u2,u,u1 are_COrte_wrt x,y by Def3; hence thesis by A1,Th18;end; hence thesis;end; theorem Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u2,v2,w,w1 are_COrtm_wrt x,y implies u,u1,u2,v2 are_COrtm_wrt x,y or v=v1 or w=w1 proof assume A1:Gen x,y;assume A2:u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u2,v2,w,w1 are_COrtm_wrt x,y; then Ortm(x,y,u),Ortm(x,y,u1) // v,v1 by Def4; then A3:v,v1 // Ortm(x,y,u),Ortm(x,y,u1) by ANALOAF:21; w,w1,v,v1 are_COrtm_wrt x,y by A1,A2,Th19; then Ortm(x,y,w),Ortm(x,y,w1) // v,v1 by Def4; then A4:v,v1 // Ortm(x,y,w),Ortm(x,y,w1) by ANALOAF:21; w,w1,u2,v2 are_COrtm_wrt x,y by A1,A2,Th19; then A5:Ortm(x,y,w),Ortm(x,y,w1) // u2,v2 by Def4; now assume A6:w<>w1 & v<>v1; then A7:Ortm(x,y,w),Ortm(x,y,w1) // Ortm(x,y,u),Ortm(x,y,u1) by A3,A4,ANALOAF:20; Ortm(x,y,w) <> Ortm(x,y,w1) by A1,A6,Th6; then Ortm(x,y,u),Ortm(x,y,u1) // u2,v2 by A5,A7,ANALOAF:20; hence thesis by Def4;end; hence thesis;end; theorem Gen x,y & u,u1,v,v1 are_COrte_wrt x,y & v,v1,w,w1 are_COrte_wrt x,y & u,u1,u2,v2 are_COrte_wrt x,y implies u2,v2,w,w1 are_COrte_wrt x,y or v=v1 or u=u1 proof assume A1:Gen x,y;assume A2:u,u1,v,v1 are_COrte_wrt x,y & v,v1,w,w1 are_COrte_wrt x,y & u,u1,u2,v2 are_COrte_wrt x,y; then A3: Orte(x,y,u),Orte(x,y,u1) // v,v1 by Def3; A4: Orte(x,y,v),Orte(x,y,v1) // w,w1 by A2,Def3; A5:Orte(x,y,u),Orte(x,y,u1) // u2,v2 by A2,Def3; now assume u<>u1 & v<>v1; then A6: Orte(x,y,u)<>Orte(x,y,u1) & Orte(x,y,v)<>Orte(x,y,v1) by A1,Th13; then v,v1 // u2,v2 by A3,A5,ANALOAF:20; then Orte(x,y,v),Orte(x,y,v1) // Orte(x,y,u2),Orte(x,y,v2) by A1,Th16; then Orte(x,y,u2),Orte(x,y,v2) // w,w1 by A4,A6,ANALOAF:20; hence thesis by Def3;end; hence thesis;end; theorem Gen x,y & u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u,u1,u2,v2 are_COrtm_wrt x,y implies u2,v2,w,w1 are_COrtm_wrt x,y or v=v1 or u=u1 proof assume A1:Gen x,y;assume A2:u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y & u,u1,u2,v2 are_COrtm_wrt x,y; then A3: Ortm(x,y,u),Ortm(x,y,u1) // v,v1 by Def4; A4: Ortm(x,y,v),Ortm(x,y,v1) // w,w1 by A2,Def4; A5:Ortm(x,y,u),Ortm(x,y,u1) // u2,v2 by A2,Def4; now assume u<>u1 & v<>v1; then A6: Ortm(x,y,u)<>Ortm(x,y,u1) & Ortm(x,y,v)<>Ortm(x,y,v1) by A1,Th6; then v,v1 // u2,v2 by A3,A5,ANALOAF:20; then Ortm(x,y,v),Ortm(x,y,v1) // Ortm(x,y,u2),Ortm(x,y,v2) by A1,Th17; then Ortm(x,y,u2),Ortm(x,y,v2) // w,w1 by A4,A6,ANALOAF:20; hence thesis by Def4;end; hence thesis;end; theorem Gen x,y implies (for v,w,u1,v1,w1 holds (not (v,v1,w,u1 are_COrte_wrt x,y) & not (v,v1,u1,w are_COrte_wrt x,y) & u1,w1,u1,w are_COrte_wrt x,y) implies ex u2 st ((v,v1,v,u2 are_COrte_wrt x,y or v,v1,u2,v are_COrte_wrt x,y) & (u1,w1,u1,u2 are_COrte_wrt x,y or u1,w1,u2,u1 are_COrte_wrt x,y))) proof assume A1:Gen x,y;let v,w,u1,v1,w1; consider u such that A2: v<>u and A3: v,v1,v,u are_COrte_wrt x,y by A1,Th40; assume not (v,v1,w,u1 are_COrte_wrt x,y) & not (v,v1,u1,w are_COrte_wrt x,y) & u1,w1,u1,w are_COrte_wrt x,y; then A4:not Orte(x,y,v),Orte(x,y,v1) // w,u1 & Orte(x,y,v),Orte(x,y,v1) // v,u & Orte(x,y,u1),Orte(x,y,w1) // u1,w & not Orte(x,y,v),Orte(x,y,v1) // u1,w by A3,Def3; then A5: v,u // Orte(x,y,v),Orte(x,y,v1) & u1,w // Orte(x,y,u1),Orte(x,y,w1) by ANALOAF:21; A6: u1<>w by A4,ANALOAF:18; A7: not (v,u // u1,w or v,u // w,u1) by A2,A4,A5,ANALOAF:20; Gen x,y implies (ex u,v st (for w ex a,b st a*u + b*v=w)) proof assume A8:Gen x,y;take x,y;thus thesis by A8,ANALMETR:def 1;end; then consider u2 such that A9: v,u // v,u2 or v,u // u2,v and A10: u1,w // u1,u2 or u1,w // u2,u1 by A1,A7,ANALOAF:31; Orte(x,y,v),Orte(x,y,v1) // v,u2 or Orte(x,y,v),Orte(x,y,v1) // u2,v by A2,A5,A9,ANALOAF:20; then A11: v,v1,v,u2 are_COrte_wrt x,y or v,v1,u2,v are_COrte_wrt x,y by Def3; Orte(x,y,u1),Orte(x,y,w1) // u1,u2 or Orte(x,y,u1),Orte(x,y,w1) // u2,u1 by A5,A6,A10,ANALOAF:20; then u1,w1,u1,u2 are_COrte_wrt x,y or u1,w1,u2,u1 are_COrte_wrt x,y by Def3; hence thesis by A11; end; theorem Gen x,y implies (ex u,v,w st (u,v,u,w are_COrte_wrt x,y & for v1,w1 st v1,w1,u,v are_COrte_wrt x,y holds (not v1,w1,u,w are_COrte_wrt x,y & not v1,w1,w,u are_COrte_wrt x,y or v1=w1))) proof assume A1:Gen x,y; Gen x,y implies ex u,v st u<>v proof assume A2:Gen x,y;take x,0.V; thus thesis by A2,Lm4;end; then consider u,v such that A3:u<>v by A1;take u,v;consider w such that A4:w<>u & u,v,u,w are_COrte_wrt x,y by A1,Th40;take w; A5:Orte(x,y,u) <> Orte(x,y,v) by A1,A3,Th13; thus u,v,u,w are_COrte_wrt x,y by A4; A6:Orte(x,y,u),Orte(x,y,v) // u,w by A4,Def3; let v1,w1; assume v1,w1,u,v are_COrte_wrt x,y; then A7:Orte(x,y,v1),Orte(x,y,w1) // u,v by Def3; now assume v1<>w1; then A8:Orte(x,y,v1)<>Orte(x,y,w1) by A1,Th13; assume v1,w1,u,w are_COrte_wrt x,y or v1,w1,w,u are_COrte_wrt x,y; then Orte(x,y,v1),Orte(x,y,w1) // u,w or Orte(x,y,v1),Orte(x,y,w1) // w,u by Def3; then u,v // u,w or u,v // w,u by A7,A8,ANALOAF:20; then Orte(x,y,u),Orte(x,y,v) // Orte(x,y,u),Orte(x,y,w) or Orte(x,y,u),Orte(x,y,v) // Orte(x,y,w),Orte(x,y,u) by A1,Th16; then u,w // Orte(x,y,u),Orte(x,y,w) or u,w // Orte(x,y,w),Orte(x,y,u) by A5,A6,ANALOAF:20; then consider a,b such that A9:a*(w-u)=b*(Orte(x,y,w)-Orte(x,y,u)) & (a<>0 or b<>0) by ANALMETR:18;take a,b; a*(w-u)=b*Orte(x,y,w-u) & (a<>0 or b<>0) by A1,A9,Th11; then a*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)=b*Orte(x,y,w-u) & (a<>0 or b<>0) by A1,Lm5; then A10:a*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= b*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y) & (a<>0 or b<>0) by Def2; A11:now assume A12:a<>0; A13:pr2(x,y,w-u)<>0 proof assume A14:not thesis; then a*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= (b*0)*x + (b*(-pr1(x,y,w-u)))*y by A10,Lm2; then (a*pr1(x,y,w-u))*x + (a*pr2(x,y,w-u))*y= (b*0)*x + (b*(-pr1(x,y,w-u)))*y by Lm2; then a*pr1(x,y,w-u)=0 by A1,Lm3; then pr1(x,y,w-u)=0 by A12,XCMPLX_1:6; then w-u=0*x + 0*y by A1,A14,Lm5 .=0.V + 0*y by RLVECT_1:23 .=0.V + 0.V by RLVECT_1:23 .=0.V by RLVECT_1:10; hence thesis by A4,RLVECT_1:35;end; (a"*a)*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= a"*(b*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y)) by A10,RLVECT_1:def 9; then (a"*a)*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= (a"*b)*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y) by RLVECT_1:def 9; then 1*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= (a"*b)*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y) by A12,XCMPLX_0:def 7; then pr1(x,y,w-u)*x + pr2(x,y,w-u)*y= (a"*b)*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y) by RLVECT_1:def 9; then pr1(x,y,w-u)*x + pr2(x,y,w-u)*y= (a"*b)*pr2(x,y,w-u)*x + (a"*b)*(-pr1(x,y,w-u))*y by Lm2; then pr1(x,y,w-u)=(a"*b)*pr2(x,y,w-u) & pr2(x,y,w-u)=(a"*b)*(-pr1(x,y,w-u)) by A1,Lm3; then pr2(x,y,w-u)=-((a"*b)*((a"*b)*pr2(x,y,w-u))) by XCMPLX_1:175; then pr2(x,y,w-u)"*(-pr2(x,y,w-u))= pr2(x,y,w-u)"*((a"*b)*((a"*b)*pr2(x,y,w-u))); then -(pr2(x,y,w-u)"*pr2(x,y,w-u))= pr2(x,y,w-u)"*((a"*b)*((a"*b)*pr2(x,y,w-u))) by XCMPLX_1:175; then -1=pr2(x,y,w-u)"*((a"*b)*((a"*b)*pr2(x,y,w-u))) by A13,XCMPLX_0:def 7 ; then -1=pr2(x,y,w-u)"*(pr2(x,y,w-u)*((a"*b)*(a"*b))) by XCMPLX_1:4; then -1=pr2(x,y,w-u)"*pr2(x,y,w-u)*((a"*b)*(a"*b)) by XCMPLX_1:4; then -1=1*((a"*b)*(a"*b)) by A13,XCMPLX_0:def 7; hence thesis by REAL_1:93;end; now assume A15:b<>0; A16:pr2(x,y,w-u)<>0 proof assume A17:not thesis; then a*(pr1(x,y,w-u)*x + 0*y)= (b*0)*x + (b*(-pr1(x,y,w-u)))*y by A10,Lm2; then (a*pr1(x,y,w-u))*x + (a*0)*y= (b*0)*x + (b*(-pr1(x,y,w-u)))*y by Lm2; then b*(-pr1(x,y,w-u))=0 by A1,Lm3; then -pr1(x,y,w-u)=0 by A15,XCMPLX_1:6; then w-u=0*x + 0*y by A1,A17,Lm5,REAL_1:26 .=0.V + 0*y by RLVECT_1:23 .=0.V + 0.V by RLVECT_1:23 .=0.V by RLVECT_1:10; hence thesis by A4,RLVECT_1:35;end; b"*(a*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y))= (b"*b)*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y) by A10,RLVECT_1:def 9; then (b"*a)*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= (b"*b)*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y) by RLVECT_1:def 9; then (b"*a)*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= 1*(pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y) by A15,XCMPLX_0:def 7; then (b"*a)*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)= pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y by RLVECT_1:def 9; then (b"*a)*pr1(x,y,w-u)*x + (b"*a)*pr2(x,y,w-u)*y= pr2(x,y,w-u)*x + (-pr1(x,y,w-u))*y by Lm2; then (b"*a)*pr1(x,y,w-u)=pr2(x,y,w-u) & (b"*a)*pr2(x,y,w-u)=-pr1(x,y,w-u) by A1,Lm3; then pr2(x,y,w-u)=(b"*a)*(-((b"*a)*pr2(x,y,w-u))); then pr2(x,y,w-u)=-((b"*a)*((b"*a)*pr2(x,y,w-u))) by XCMPLX_1:175; then pr2(x,y,w-u)"*(-pr2(x,y,w-u))= pr2(x,y,w-u)"*((b"*a)*((b"*a)*pr2(x,y,w-u))); then -(pr2(x,y,w-u)"*pr2(x,y,w-u))= pr2(x,y,w-u)"*((b"*a)*((b"*a)*pr2(x,y,w-u))) by XCMPLX_1:175; then -1=pr2(x,y,w-u)"*((b"*a)*((b"*a)*pr2(x,y,w-u))) by A16,XCMPLX_0:def 7 ; then -1=pr2(x,y,w-u)"*(pr2(x,y,w-u)*((b"*a)*(b"*a))) by XCMPLX_1:4; then -1=pr2(x,y,w-u)"*pr2(x,y,w-u)*((b"*a)*(b"*a)) by XCMPLX_1:4; then -1=1*((b"*a)*(b"*a)) by A16,XCMPLX_0:def 7; hence thesis by REAL_1:93;end; hence thesis by A9,A11;end; hence thesis;end; theorem Gen x,y implies (for v,w,u1,v1,w1 holds (not v,v1,w,u1 are_COrtm_wrt x,y & not v,v1,u1,w are_COrtm_wrt x,y & u1,w1,u1,w are_COrtm_wrt x,y) implies ex u2 st ((v,v1,v,u2 are_COrtm_wrt x,y or v,v1,u2,v are_COrtm_wrt x,y) & (u1,w1,u1,u2 are_COrtm_wrt x,y or u1,w1,u2,u1 are_COrtm_wrt x,y))) proof assume A1:Gen x,y;let v,w,u1,v1,w1; consider u such that A2: v<>u and A3: v,v1,v,u are_COrtm_wrt x,y by A1,Th41; assume not (v,v1,w,u1 are_COrtm_wrt x,y) & not (v,v1,u1,w are_COrtm_wrt x,y) & u1,w1,u1,w are_COrtm_wrt x,y; then A4:not Ortm(x,y,v),Ortm(x,y,v1) // w,u1 & Ortm(x,y,v),Ortm(x,y,v1) // v,u & Ortm(x,y,u1),Ortm(x,y,w1) // u1,w & not Ortm(x,y,v),Ortm(x,y,v1) // u1,w by A3,Def4; then A5: v,u // Ortm(x,y,v),Ortm(x,y,v1) & u1,w // Ortm(x,y,u1),Ortm(x,y,w1) by ANALOAF:21; A6: u1<>w by A4,ANALOAF:18; A7: not (v,u // u1,w or v,u // w,u1) by A2,A4,A5,ANALOAF:20; Gen x,y implies (ex u,v st (for w ex a,b st a*u + b*v=w)) proof assume A8:Gen x,y;take x,y;thus thesis by A8,ANALMETR:def 1;end; then consider u2 such that A9: v,u // v,u2 or v,u // u2,v and A10: u1,w // u1,u2 or u1,w // u2,u1 by A1,A7,ANALOAF:31; Ortm(x,y,v),Ortm(x,y,v1) // v,u2 or Ortm(x,y,v),Ortm(x,y,v1) // u2,v by A2,A5,A9,ANALOAF:20; then A11: v,v1,v,u2 are_COrtm_wrt x,y or v,v1,u2,v are_COrtm_wrt x,y by Def4; Ortm(x,y,u1),Ortm(x,y,w1) // u1,u2 or Ortm(x,y,u1),Ortm(x,y,w1) // u2,u1 by A5,A6,A10,ANALOAF:20; then u1,w1,u1,u2 are_COrtm_wrt x,y or u1,w1,u2,u1 are_COrtm_wrt x,y by Def4; hence thesis by A11; end; theorem Gen x,y implies (ex u,v,w st (u,v,u,w are_COrtm_wrt x,y & for v1,w1 holds (v1,w1,u,v are_COrtm_wrt x,y implies (not v1,w1,u,w are_COrtm_wrt x,y & not v1,w1,w,u are_COrtm_wrt x,y or v1=w1)))) proof assume A1:Gen x,y;take u=0*x+0*y,v=1*x+1*y,w=1*x+(-1)*y; A2:pr1(x,y,u)=0 & pr2(x,y,u)=0 & pr1(x,y,v)=1 & pr2(x,y,v)=1 by A1,Lm6; then A3:Ortm(x,y,u)=0*x+0*y by Def1,REAL_1:26; Ortm(x,y,v)=1*x+(-1)*y by A2,Def1; then A4: Ortm(x,y,u),Ortm(x,y,v) // u,w by A3,ANALOAF:17; for v1,w1 holds (v1,w1,u,v are_COrtm_wrt x,y implies (not v1,w1,u,w are_COrtm_wrt x,y & not v1,w1,w,u are_COrtm_wrt x,y or v1=w1)) proof let v1,w1; assume v1,w1,u,v are_COrtm_wrt x,y; then A5:Ortm(x,y,v1),Ortm(x,y,w1) // u,v by Def4; now assume v1<>w1; then A6:Ortm(x,y,v1)<>Ortm(x,y,w1) by A1,Th6; assume v1,w1,u,w are_COrtm_wrt x,y or v1,w1,w,u are_COrtm_wrt x,y; then Ortm(x,y,v1),Ortm(x,y,w1) // u,w or Ortm(x,y,v1),Ortm(x,y,w1) // w,u by Def4; then u,v // u,w or u,v // w,u by A5,A6,ANALOAF:20; then consider a,b such that A7:a*(v-u)=b*(w-u) & (a<>0 or b<>0) by ANALMETR:18;take a,b; u=0.V+0*y by RLVECT_1:23 .=0.V+0.V by RLVECT_1:23 .=0.V by RLVECT_1:10; then a*v=b*(w-0.V) & (a<>0 or b<>0) by A7,RLVECT_1:26; then A8:a*v=b*w & (a<>0 or b<>0) by RLVECT_1:26; A9:now assume A10:a<>0; a"*a*v=a"*(b*w) by A8,RLVECT_1:def 9; then a"*a*v=a"*b*w by RLVECT_1:def 9; then 1*v=a"*b*w by A10,XCMPLX_0:def 7; then 1*v=a"*b*1*x+a"*b*(-1)*y by Lm2; then 1*1*x+1*1*y=a"*b*1*x+a"*b*(-1)*y by Lm2; then a*1=a*(a"*(b*1)) & 1=a"*b*(-1) by A1,Lm3; then a*1=a*a"*(b*1) & 1=a"*b*(-1) by XCMPLX_1:4; then 1=a"*a*(-1) by A10,XCMPLX_0:def 7; then 1=1*(-1) by A10,XCMPLX_0:def 7; hence thesis;end; now assume A11:b<>0; b"*a*v=b"*(b*w) by A8,RLVECT_1:def 9; then b"*a*v=b"*b*w by RLVECT_1:def 9; then b"*a*v=1*w by A11,XCMPLX_0:def 7; then b"*a*1*x+b"*a*1*y=1*w by Lm2; then b"*a*1*x+b"*a*1*y=1*1*x+1*(-1)*y by Lm2; then b*1=b*(b"*(a*1)) & -1=b"*a*1 by A1,Lm3; then b*1=b*b"*(a*1) & -1=b"*a*1 by XCMPLX_1:4; then -1=b"*b*1 by A11,XCMPLX_0:def 7; hence thesis by A11,XCMPLX_0:def 7;end; hence thesis by A7,A9;end;hence thesis;end; hence thesis by A4,Def4;end; reserve uu,vv for set; definition let V;let x,y; func CORTE(V,x,y) -> Relation of [:the carrier of V,the carrier of V:] means :Def5: [uu,vv] in it iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y; existence proof set VV = [:the carrier of V,the carrier of V :]; defpred P[set,set] means ex u1,u2,v1,v2 st $1=[u1,u2] & $2=[v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y; consider P being Relation of VV,VV such that A1: for uu,vv holds ([uu,vv] in P iff uu in VV & vv in VV & P[uu,vv]) from Rel_On_Set_Ex; take P;let uu,vv; thus [uu,vv] in P implies ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y by A1; assume A2:ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y; then uu in VV & vv in VV by ZFMISC_1:def 2; hence [uu,vv] in P by A1,A2; end; uniqueness proof let P,Q be Relation of [:the carrier of V,the carrier of V:] such that A3:[uu,vv] in P iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y and A4:[uu,vv] in Q iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y; for uu,vv holds [uu,vv] in P iff [uu,vv] in Q proof let uu,vv; [uu,vv] in P iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrte_wrt x,y by A3; hence thesis by A4;end; hence thesis by RELAT_1:def 2;end; end; definition let V;let x,y; func CORTM(V,x,y) -> Relation of [:the carrier of V,the carrier of V:] means :Def6: [uu,vv] in it iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y; existence proof set VV = [:the carrier of V,the carrier of V :]; defpred P[set,set] means ex u1,u2,v1,v2 st $1=[u1,u2] & $2=[v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y; consider P being Relation of VV,VV such that A1:for uu,vv holds ([uu,vv] in P iff uu in VV & vv in VV & P[uu,vv]) from Rel_On_Set_Ex; take P;let uu,vv; thus [uu,vv] in P implies ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y by A1; assume A2:ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y; then uu in VV & vv in VV by ZFMISC_1:def 2; hence [uu,vv] in P by A1,A2; end; uniqueness proof let P,Q be Relation of [:the carrier of V,the carrier of V:] such that A3:[uu,vv] in P iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y and A4:[uu,vv] in Q iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y; for uu,vv holds [uu,vv] in P iff [uu,vv] in Q proof let uu,vv; [uu,vv] in P iff ex u1,u2,v1,v2 st uu=[u1,u2] & vv=[v1,v2] & u1,u2,v1,v2 are_COrtm_wrt x,y by A3; hence thesis by A4;end; hence thesis by RELAT_1:def 2;end; end; definition let V;let x,y; func CESpace(V,x,y) -> strict AffinStruct equals :Def7: AffinStruct (# the carrier of V,CORTE(V,x,y) #); correctness; end; definition let V;let x,y; cluster CESpace(V,x,y) -> non empty; coherence proof CESpace(V,x,y) = AffinStruct (# the carrier of V,CORTE(V,x,y) #) by Def7; hence the carrier of CESpace(V,x,y) is non empty; end; end; definition let V;let x,y; func CMSpace(V,x,y) -> strict AffinStruct equals :Def8: AffinStruct (# the carrier of V,CORTM(V,x,y) #); correctness; end; definition let V;let x,y; cluster CMSpace(V,x,y) -> non empty; coherence proof CMSpace(V,x,y) = AffinStruct (# the carrier of V,CORTM(V,x,y) #) by Def8; hence the carrier of CMSpace(V,x,y) is non empty; end; end; theorem uu is Element of CESpace(V,x,y) iff uu is VECTOR of V proof A1:uu is Element of CESpace(V,x,y) implies uu is VECTOR of V proof assume uu is Element of CESpace(V,x,y); then uu is Element of AffinStruct (# the carrier of V,CORTE(V,x,y) #) by Def7; hence uu is VECTOR of V;end; uu is VECTOR of V implies uu is Element of CESpace(V,x,y) proof assume uu is VECTOR of V; then uu is Element of AffinStruct (# the carrier of V,CORTE(V,x,y) #); hence thesis by Def7;end; hence thesis by A1;end; theorem uu is Element of CMSpace(V,x,y) iff uu is VECTOR of V proof A1:uu is Element of CMSpace(V,x,y) implies uu is VECTOR of V proof assume uu is Element of CMSpace(V,x,y); then uu is Element of AffinStruct (# the carrier of V,CORTM(V,x,y) #) by Def8; hence uu is VECTOR of V;end; uu is VECTOR of V implies uu is Element of CMSpace(V,x,y) proof assume uu is VECTOR of V; then uu is Element of AffinStruct (# the carrier of V,CORTM(V,x,y) #); hence thesis by Def8;end; hence thesis by A1;end; reserve p,q,r,s for Element of CESpace(V,x,y); theorem u=p & v=q & u1=r & v1=s implies (p,q // r,s iff u,v,u1,v1 are_COrte_wrt x,y) proof assume A1: u=p & v=q & u1=r & v1=s; A2:p,q // r,s implies u,v,u1,v1 are_COrte_wrt x,y proof assume p,q // r,s; then [[p,q],[r,s]] in the CONGR of CESpace(V,x,y) by ANALOAF:def 2; then [[u,v],[u1,v1]] in the CONGR of AffinStruct (# the carrier of V,CORTE(V,x,y) #) by A1,Def7; then consider u1',u2',v1',v2' being VECTOR of V such that A3: [u,v]=[u1',u2'] & [u1,v1]=[v1',v2'] and A4: u1',u2',v1',v2' are_COrte_wrt x,y by Def5; u=u1' & v=u2' & u1=v1' & v1=v2' by A3,ZFMISC_1:33; hence thesis by A4;end; u,v,u1,v1 are_COrte_wrt x,y implies p,q // r,s proof assume u,v,u1,v1 are_COrte_wrt x,y; then [[u,v],[u1,v1]] in the CONGR of AffinStruct (# the carrier of V,CORTE(V,x,y) #) by Def5; then [[p,q],[r,s]] in the CONGR of CESpace(V,x,y) by A1,Def7; hence thesis by ANALOAF:def 2;end; hence thesis by A2;end; reserve p,q,r,s for Element of CMSpace(V,x,y); theorem u=p & v=q & u1=r & v1=s implies (p,q // r,s iff u,v,u1,v1 are_COrtm_wrt x,y) proof assume A1: u=p & v=q & u1=r & v1=s; A2:p,q // r,s implies u,v,u1,v1 are_COrtm_wrt x,y proof assume p,q // r,s; then [[p,q],[r,s]] in the CONGR of CMSpace(V,x,y) by ANALOAF:def 2; then [[u,v],[u1,v1]] in the CONGR of AffinStruct (# the carrier of V,CORTM(V,x,y) #) by A1,Def8; then consider u1',u2',v1',v2' being VECTOR of V such that A3: [u,v]=[u1',u2'] & [u1,v1]=[v1',v2'] and A4: u1',u2',v1',v2' are_COrtm_wrt x,y by Def6; u=u1' & v=u2' & u1=v1' & v1=v2' by A3,ZFMISC_1:33; hence thesis by A4;end; u,v,u1,v1 are_COrtm_wrt x,y implies p,q // r,s proof assume u,v,u1,v1 are_COrtm_wrt x,y; then [[u,v],[u1,v1]] in the CONGR of AffinStruct (# the carrier of V,CORTM(V,x,y) #) by Def6; then [[p,q],[r,s]] in the CONGR of CMSpace(V,x,y) by A1,Def8; hence thesis by ANALOAF:def 2;end; hence thesis by A2;end;