theorem
  a,b are_isomorphic iff Hom(a,b)<>{} & Hom(b,a)<>{} & ex f,f9 st
  f*f9 = id b & f9*f = id a
proof
  thus a,b are_isomorphic implies Hom(a,b)<>{} & Hom(b,a)<>{} & ex f,f9 st f*
  f9 = id b & f9*f = id a
  proof
    given f such that
A1: f is invertible;
    thus Hom(a,b) <> {} & Hom(b,a) <> {} by A1;
    take f;
    thus thesis by A1;
  end;
  assume that
A2: Hom(a,b)<>{} and
A3: Hom(b,a)<>{};
  given f such that
A4: ex f9 st f*f9 = id b & f9*f = id a;
  take f;
  thus thesis by A2,A3,A4;
end;
