
theorem Th28:
  for C being non empty category,
      a,b being Object of C, f being Morphism of a,b
  st f is section_ holds f is monomorphism
  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 section_;
    then consider g be Morphism of b,a such that
A2: g * f = id- a;
    thus Hom(a,b) <> {} by A1;
    let c be Object of C;
    assume
A3: Hom(c,a) <> {};
    let g1,g2 be Morphism of c,a;
    assume
A4: f * g1 = f * g2;
A5: (g * f) * g1 = g * (f * g1) by A1,A3,Th23
    .= (g * f) * g2 by A1,A4,A3,Th23;
    thus g1 = (g * f) * g1 by A3,A2,Th18
    .= g2 by A5,A3,A2,Th18;
  end;
