reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;

theorem Th13:
  u,y // y,v implies u#y,y // y,y#v
proof
  set p=u#y, q=y#v;
  u,p // p,y by Th11;
  then p,y // u,y by Lm1;
  then
A1: u,y // p,y by ANALOAF:12;
  y,q // q,v by Th11;
  then y,q // y,v by Lm1;
  then
A2: y,v // y,q by ANALOAF:12;
  assume
A3: u,y // y,v;
  now
    assume that
A4: u<>y and
A5: y<>v;
    y,v // p,y by A3,A1,A4,ANALOAF:11;
    hence thesis by A2,A5,ANALOAF:11;
  end;
  hence thesis by ANALOAF:9;
end;
