
theorem
  for C being non empty category, a,b being Object of C
  holds b is initial & b,a are_isomorphic implies a is initial
proof
  let C be non empty category, a,b be Object of C;
  assume
A1: b is initial;
  assume b,a are_isomorphic;
  then consider f be Morphism of b,a such that
A2: f is isomorphism by CAT_7:def 10;
A3: Hom(a,b) <> {} by A2,CAT_7:def 9;
  let c be Object of C;
  consider h being Morphism of b,c such that
A4: for g being Morphism of b,c holds h = g by A1;
  Hom(b,c) <> {} by A1;
  hence
A5: Hom(a,c) <> {} by A3,CAT_7:22;
  consider f1 be Morphism of a,b such that
  f1*f = id- b and
A6: f*f1 = id- a by A2,CAT_7:def 9;
A7: Hom(b,a) <> {} by A2,CAT_7:def 9;
  take h*f1;
  let h1 be Morphism of a,c;
  thus h*f1 = (h1*f)*f1 by A4
    .= h1*(f*f1) by A3,A5,A7,CAT_7:23
    .= h1 by A6,A5,CAT_7:18;
end;
