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 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:9;
    then
A4: u,v,w,u1 are_COrte_wrt x,y;
    now
      assume w=u1;
      then x=0.V by RLVECT_1:9;
      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:16;
    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:12;
    then Orte(x,y,u1),Orte(x,y,w) // u,v by ANALOAF:12;
    then
A8: u1,w,u,v are_COrte_wrt x,y;
    now
      assume w=u1;
      then w= w+(u2-v2) by RLVECT_1:def 3;
      then u2-v2=0.V by RLVECT_1:9;
      hence contradiction by A5,A6,A7,RLVECT_1:21;
    end;
    hence thesis by A1,A8,Th18;
  end;
  hence thesis by A2;
end;
