
theorem Th99:
  for G1, G2 being _Graph
  for F being PGraphMapping of G1, G2, H being Subgraph of G2
  holds F_E"the_Edges_of H c= G1.edgesBetween(F_V"the_Vertices_of H)
proof
  let G1, G2 be _Graph;
  let F be PGraphMapping of G1, G2, H be Subgraph of G2;
  now
    let x be object;
    assume x in F_E"the_Edges_of H;
    then A1: x in dom F_E & F_E.x in the_Edges_of H by FUNCT_1:def 7;
    set v = (the_Source_of G1).x, w = (the_Target_of G1).x;
    A2: v in dom F_V & w in dom F_V by A1, GLIB_010:5;
    A3: x Joins v,w,G1 by A1, GLIB_000:def 13;
    then F_E.x Joins F_V.v,F_V.w,G2 by A1, A2, GLIB_010:4;
    then F_E.x Joins F_V.v,F_V.w,H by A1, GLIB_000:73;
    then F_V.v in the_Vertices_of H & F_V.w in the_Vertices_of H
      by GLIB_000:13;
    then v in F_V"the_Vertices_of H & w in F_V"the_Vertices_of H
      by A2, FUNCT_1:def 7;
    hence x in G1.edgesBetween(F_V"the_Vertices_of H) by A3, GLIB_000:32;
  end;
  hence thesis by TARSKI:def 3;
end;
