theorem Th41:
  f, g form_embedding_of C1, C2 implies
  dom f = the carrier of G1 & rng f c= the carrier of G2 &
  dom g = the carrier' of G1 & rng g c= the carrier' of G2
proof
  assume that f is one-to-one and g is one-to-one and
A1: dom f = the carrier of G1 and
A2: dom g = the carrier' of G1 and
A3: rng f c= the carrier of G2 and
A4: rng g c= the carrier' of G2;
  thus thesis by A1,A2,A3,A4;
end;
