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