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 & 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 that
A2: u,u1,v,v1 are_COrte_wrt x,y and
A3: v,v1,w,w1 are_COrte_wrt x,y and
A4: u2,v2,w,w1 are_COrte_wrt x,y;
  v,v1,u1,u are_COrte_wrt x,y by A1,A2,Th18;
  then
A5: Orte(x,y,v),Orte(x,y,v1) // u1,u;
  Orte(x,y,v),Orte(x,y,v1) // w,w1 by A3;
  then
A6: w,w1 // Orte(x,y,v),Orte(x,y,v1) by ANALOAF:12;
  Orte(x,y,u2),Orte(x,y,v2) // w,w1 by A4;
  then
A7: w,w1 // Orte(x,y,u2),Orte(x,y,v2) by ANALOAF:12;
  now
    assume that
A8: w<>w1 and
A9: v<>v1;
    Orte(x,y,v),Orte(x,y,v1) // Orte(x,y,u2),Orte(x,y,v2)
    by A6,A7,A8,ANALOAF:11;
    then Orte(x,y,u2),Orte(x,y,v2) // u1,u by A1,A5,A9,Th13,ANALOAF:11;
    then Orte(x,y,v2),Orte(x,y,u2) // u,u1 by ANALOAF:12;
    then v2,u2,u,u1 are_COrte_wrt x,y;
    hence thesis by A1,Th18;
  end;
  hence thesis;
end;
