
theorem
  for C being category, a,b,c being Object of C,
      f1 being Morphism of a,b, f2 being Morphism of b,c
  st f1 is epimorphism & f2 is epimorphism
  holds f2 * f1 is epimorphism
  proof
    let C be category;
    let a,b,c be Object of C;
    let f1 be Morphism of a,b;
    let f2 be Morphism of b,c;
    assume
A1: f1 is epimorphism;
    assume
A2: f2 is epimorphism;
    hence Hom(a,c) <> {} by A1,Th22;
    let d be Object of C;
    assume
A3: Hom(c,d) <> {};
    let g1,g2 be Morphism of c,d;
    assume
A4: g1 * (f2 * f1) = g2 * (f2 * f1);
A5: Hom(b,d) <> {} by A3,A2,Th22;
    (g1 * f2) * f1 = g1 * (f2 * f1) by A1,A2,A3,Th23
    .= (g2 * f2) * f1 by A4,A1,A2,A3,Th23;
    hence g1 = g2 by A3,A2,A1,A5;
  end;
