reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;

theorem
  x1 <> x2 implies ((x1,x2)-->(p1,p2))"*"((x1,x2)-->(q1,q2)) = (x1,x2)
  -->(p1(*)q1,p2(*)q2)
proof
  set F1 = (x1,x2)-->(p1,p2),
      F2 = (x1,x2)-->(q1,q2), G = (x1,x2)-->(p1(*)q1,p2(*)
  q2);
  assume
A1: x1 <> x2;
  now
    let x;
    assume
A2: x in {x1,x2};
    then x = x1 or x = x2 by TARSKI:def 2;
    then
    F1/.x = p1 & F2/.x = q1 & G/.x = p1(*)q1 or F1/.x = p2 & F2/.x = q2 & G
    /.x = p2(*)q2 by A1,Th3;
    hence (F1"*"F2)/.x = G/.x by A2,Def7;
  end;
  hence thesis by Th1;
end;
