
theorem
  for G1, G2 being WGraph, H being WSubgraph of G2,
    F being PGraphMapping of G1, G2
  st F is weight-preserving holds H |` F is weight-preserving
proof
  let G1, G2 be WGraph, H being WSubgraph of G2;
  let F be PGraphMapping of G1, G2;
  assume A1: F is weight-preserving;
  (the_Edges_of H)|`F_E c= F_E by RELAT_1:86;
  then A2: dom ((the_Edges_of H)|`F_E) /\  dom F_E = dom (H|`F)_E
    by XBOOLE_1:28, RELAT_1:11;
  the_Weight_of H * (H|`F)_E
     = ((the_Weight_of G2) | the_Edges_of H) * (H|`F)_E by GLIB_003:def 10
    .= the_Weight_of G2 * ((the_Edges_of H)|`((the_Edges_of H)|`F_E))
      by GROUP_9:121
    .= the_Weight_of G2 * (F_E | dom((the_Edges_of H)|`F_E)) by GLIB_009:4
    .= ((the_Weight_of G1) | dom F_E) | dom (H|`F)_E by A1, RELAT_1:83
    .= (the_Weight_of G1) | dom (H|`F)_E by A2, RELAT_1:71;
  hence thesis;
end;
