
theorem
  for C being composable with_identities CategoryStr,
      a,b being Object of C, f1 being Morphism of a,b
  st Hom(a,b) <> {} & f1 is identity holds f1 is epimorphism
  proof
    let C be composable with_identities CategoryStr;
    let a,b be Object of C;
    let f1 be Morphism of a,b;
    assume
A1: Hom(a,b) <> {};
    assume
A2: f1 is identity;
    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 * f1 = g2 * f1;
    thus g1 = g1 * f1 by A3,A1,A2,Th24
    .= g2 by A3,A1,A4,A2,Th24;
  end;
