
theorem
  for G1, G2 being VGraph, H being VSubgraph of G2,
    F being PGraphMapping of G1, G2
  st F is vlabel-preserving holds H |` F is vlabel-preserving
proof
  let G1, G2 be VGraph, H being VSubgraph of G2;
  let F be PGraphMapping of G1, G2;
  assume A1: F is vlabel-preserving;
  (the_Vertices_of H)|`F_V c= F_V by RELAT_1:86;
  then A2: dom ((the_Vertices_of H)|`F_V) /\  dom F_V = dom (H|`F)_V
    by XBOOLE_1:28, RELAT_1:11;
  the_VLabel_of H * (H|`F)_V
     = ((the_VLabel_of G2) | the_Vertices_of H) * (H|`F)_V by GLIB_003:def 12
    .= the_VLabel_of G2 * ((the_Vertices_of H)|`((the_Vertices_of H)|`F_V))
      by GROUP_9:121
    .= the_VLabel_of G2 * (F_V | dom((the_Vertices_of H)|`F_V)) by GLIB_009:4
    .= ((the_VLabel_of G1) | dom F_V) | dom (H|`F)_V by A1, RELAT_1:83
    .= (the_VLabel_of G1) | dom (H|`F)_V by A2, RELAT_1:71;
  hence thesis;
end;
