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_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 by A1,Lm6;
A3: pr2(x,y,u)=0 by A1,Lm6;
A4: pr1(x,y,v)=1 by A1,Lm6;
  pr2(x,y,v)=1 by A1,Lm6;
  then
A5: Ortm(x,y,u),Ortm(x,y,v) // u,w by A2,A3,A4,ANALOAF:8;
  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
A6: Ortm(x,y,v1),Ortm(x,y,w1) // u,v;
    now
      assume
A7:   v1<>w1;
      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;
      then u,v // u,w or u,v // w,u by A1,A6,A7,Th6,ANALOAF:11;
      then consider a,b such that
A8:   a*(v-u)=b*(w-u) and
A9:   a<>0 or b<>0 by ANALMETR:14;
      take a,b;
      u=0.V+0*y by RLVECT_1:10
        .=0.V+0.V by RLVECT_1:10
        .=0.V by RLVECT_1:4;
      then a*v=b*(w-0.V) by A8,RLVECT_1:13;
      then
A10:  a*v=b*w by RLVECT_1:13;
A11:  now
        assume
A12:    a<>0;
        a"*a*v=a"*(b*w) by A10,RLVECT_1:def 7;
        then a"*a*v=a"*b*w by RLVECT_1:def 7;
        then 1*v=a"*b*w by A12,XCMPLX_0:def 7;
        then 1*v=a"*b*1*x+a"*b*(-1)*y by Lm2;
        then
A13:    1*1*x+1*1*y=a"*b*1*x+a"*b*(-1)*y by Lm2;
        then a*1=a*(a"*(b*1)) by A1,Lm3;
        then
A14:    a*1=a*a"*(b*1);
        1=a"*b*(-1) by A1,A13,Lm3;
        then 1=a"*a*(-1) by A12,A14,XCMPLX_0:def 7;
        hence thesis by A12,XCMPLX_0:def 7;
      end;
      now
        assume
A15:    b<>0;
        b"*a*v=b"*(b*w) by A10,RLVECT_1:def 7;
        then b"*a*v=b"*b*w by RLVECT_1:def 7;
        then b"*a*v=1*w by A15,XCMPLX_0:def 7;
        then b"*a*1*x+b"*a*1*y=1*w by Lm2;
        then
A16:    b"*a*1*x+b"*a*1*y=1*1*x+1*(-1)*y by Lm2;
        then b*1=b*(b"*(a*1)) by A1,Lm3;
        then
A17:    b*1=b*b"*(a*1);
        -1=b"*a*1 by A1,A16,Lm3;
        then -1=b"*b*1 by A15,A17,XCMPLX_0:def 7;
        hence thesis by A15,XCMPLX_0:def 7;
      end;
      hence thesis by A9,A11;
    end;
    hence thesis;
  end;
  hence thesis by A5;
end;
