theorem
  f is retraction & f is monic implies f is invertible
proof assume
A1: Hom(a,b) <> {} & Hom(b,a) <> {};
  given i being Morphism of b,a such that
A2: f*i = id b;
  assume
A3: f is monic;
  thus Hom(a,b) <> {} & Hom(b,a) <> {} by A1;
  take i;
  thus f*i = id b by A2;
A4: f*(i*f) = (id b)*f by A1,A2,CAT_1:25
    .= f by A1,CAT_1:28
    .= f*id a by A1,CAT_1:29;
   Hom(a,a) <> {};
  hence i*f = id a by A3,A4;
end;
