
theorem Th32:
  for C being non empty category,
      a,b being Object of C, f being Morphism of a,b
  st f is retraction holds f is epimorphism
  proof
    let C be non empty category;
    let a,b be Object of C;
    let f be Morphism of a,b;
    assume
A1: f is retraction;
    then consider g be Morphism of b,a such that
A2: f * g = id- b;
    thus Hom(a,b) <> {} by A1;
    let c be Object of C;
    assume
A3: Hom(b,c) <> {};
    let g1,g2 be Morphism of b,c;
    assume
A4: g1 * f = g2 * f;
A5: g1 * (f * g) = (g1 * f) * g by A1,A3,Th23
    .= g2 * (f * g) by A1,A4,A3,Th23;
    thus g1 = g1 * (f * g) by A3,A2,Th18
    .= g2 by A5,A3,A2,Th18;
  end;
