
theorem
  for G1, G2 being WGraph, H being WSubgraph of G1,
    F being PGraphMapping of G1, G2
  st F is weight-preserving holds F | H is weight-preserving
proof
  let G1, G2 be WGraph, H being WSubgraph of G1;
  let F be PGraphMapping of G1, G2;
  assume A1: F is weight-preserving;
  the_Weight_of G2 * (F|H)_E
     = (the_Weight_of G2 * F_E) | the_Edges_of H by RELAT_1:83
    .= (the_Weight_of G1) | (dom F_E /\ (the_Edges_of H /\ the_Edges_of H))
      by A1, RELAT_1:71
    .= (the_Weight_of G1) | (the_Edges_of H /\ (dom F_E /\ the_Edges_of H))
      by XBOOLE_1:16
    .= (the_Weight_of G1) | (the_Edges_of H /\ dom (F|H)_E) by Th59
    .= ((the_Weight_of G1) | the_Edges_of H) | dom (F|H)_E by RELAT_1:71
    .= (the_Weight_of H) | dom (F|H)_E by GLIB_003:def 10;
  hence thesis;
end;
