
theorem Th53:
  for C1,C2 being non empty category, f1,g1 being morphism of C1,
      f2,g2 being morphism of C2
  st [f1,f2] = [g1,g2] holds f1 = g1 & f2 = g2
  proof
    let C1,C2 be non empty category;
    let f1,g1 be morphism of C1;
    let f2,g2 be morphism of C2;
    assume
A1: [f1,f2] = [g1,g2];
    pr1(C1,C2).[f1,f2] = f1 & pr1(C1,C2).[g1,g2] = g1 by Def23;
    hence f1 = g1 by A1;
    pr2(C1,C2).[f1,f2] = f2 & pr2(C1,C2).[g1,g2] = g2 by Def23;
    hence f2 = g2 by A1;
  end;
