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