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;

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 and
A5: u,v,u,w are_COrte_wrt x,y by A1,Th40;
  take w;
  thus u,v,u,w are_COrte_wrt x,y by A5;
A6: Orte(x,y,u),Orte(x,y,v) // u,w by A5;
  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;
  now
    assume
A8: v1<>w1;
    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;
    then u,v // u,w or u,v // w,u by A1,A7,A8,Th13,ANALOAF:11;
    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 A1,A3,A6,Th13,ANALOAF:11;
    then consider a,b such that
A9: a*(w-u)=b*(Orte(x,y,w)-Orte(x,y,u)) and
A10: a<>0 or b<>0 by ANALMETR:14;
    take a,b;
    a*(w-u)=b*Orte(x,y,w-u) by A1,A9,Th11;
    then
A11: a*(pr1(x,y,w-u)*x + pr2(x,y,w-u)*y)=b*Orte(x,y,w-u) by A1,Lm5;
A12: now
      assume
A13:  a<>0;
A14:  pr2(x,y,w-u)<>0
      proof
        assume
A15:    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 A11,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 A13,XCMPLX_1:6;
        then w-u=0*x + 0*y by A1,A15,Lm5
          .=0.V + 0*y by RLVECT_1:10
          .=0.V + 0.V by RLVECT_1:10
          .=0.V by RLVECT_1:4;
        hence thesis by A4,RLVECT_1:21;
      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 A11,RLVECT_1:def 7;
      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 7;
      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 A13,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 8;
      then
A16:  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) by A1,Lm3;
      then
A17:  pr2(x,y,w-u)=-((a"*b)*((a"*b)*pr2(x,y,w-u))) by A1,A16,Lm3;
      -(pr2(x,y,w-u)"*pr2(x,y,w-u)) = (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 A17;
      then -1=pr2(x,y,w-u)"*pr2(x,y,w-u)*((a"*b)*(a"*b)) by A14,XCMPLX_0:def 7;
      then -1=1*((a"*b)*(a"*b)) by A14,XCMPLX_0:def 7;
      hence thesis by XREAL_1:63;
    end;
    now
      assume
A18:  b<>0;
A19:  pr2(x,y,w-u)<>0
      proof
        assume
A20:    not thesis;
        then a*(pr1(x,y,w-u)*x + 0*y)=
        (b*0)*x + (b*(-pr1(x,y,w-u)))*y by A11,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 A18,XCMPLX_1:6;
        then w-u=0*x + (-0)*y by A1,A20,Lm5
          .=0.V + 0*y by RLVECT_1:10
          .=0.V + 0.V by RLVECT_1:10
          .=0.V by RLVECT_1:4;
        hence thesis by A4,RLVECT_1:21;
      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 A11,RLVECT_1:def 7;
      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 7;
      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 A18,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 8;
      then
A21:  (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)*pr2(x,y,w-u)=-pr1(x,y,w-u) by A1,Lm3;
      then
A22:  pr2(x,y,w-u)=(b"*a)*(-((b"*a)*pr2(x,y,w-u))) by A1,A21,Lm3;
      -(pr2(x,y,w-u)"*pr2(x,y,w-u)) = (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 A22;
      then -1=pr2(x,y,w-u)"*pr2(x,y,w-u)*((b"*a)*(b"*a)) by A19,XCMPLX_0:def 7;
      then -1=1*((b"*a)*(b"*a)) by A19,XCMPLX_0:def 7;
      hence thesis by XREAL_1:63;
    end;
    hence thesis by A10,A12;
  end;
  hence thesis;
end;
