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 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;
  then v,v1 // Orte(x,y,u),Orte(x,y,u1) by ANALOAF:12;
  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:12;
  -u1-(-u)=u+(-u1) by RLVECT_1:17
    .=u-u1;
  then
A3: -u,-u1 // u1,u by ANALOAF:15;
A4: -u<>-u1 implies thesis by A2,A3,ANALOAF:11;
  now
    assume -u=-u1;
    then u=-(-u1) by RLVECT_1:17
      .= u1 by RLVECT_1:17;
    then Orte(x,y,v),Orte(x,y,v1) // u1,u by ANALOAF:9;
    hence thesis;
  end;
  hence thesis by A4;
end;
