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