reserve G, G1, G2 for _Graph, H for Subgraph of G;

theorem Th50:
  for F being PGraphMapping of G1, G2 st F is total onto
  holds G2.allInducedSG() c= rng(SG2SGFunc(F) | G1.allInducedSG())
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is total onto;
  set f = SG2SGFunc(F) | G1.allInducedSG();
  A2: dom f = G1.allInducedSG() by FUNCT_2:def 1;
  now
    let x be object;
    assume A3: x in G2.allInducedSG();
    A4: rng F == G2 by A1, GLIB_010:56;
    then x in (rng F).allInducedSG() by A3, Th49;
    then consider V2 being non empty Subset of the_Vertices_of rng F
      such that A5: x = the plain inducedSubgraph of rng F,V2;
    reconsider H2 = x as plain inducedSubgraph of rng F,V2 by A5;
    set H1 = the plain inducedSubgraph of G1, F_V"the_Vertices_of H2;
    reconsider y = H1 as object;
    rng F_V = the_Vertices_of G2 by A1, GLIB_010:def 12
      .= the_Vertices_of rng F by A4, GLIB_000:def 34;
    then A6: F_V"V2 <> {} by RELAT_1:139;
    the_Vertices_of H2 = V2 by GLIB_000:def 37;
    then A7: H1 in G1.allInducedSG() by A6;
    then A8: y in dom f by A2;
    A9: (SG2SGFunc F).H1 = rng(F | H1) by Def5;
    dom F_E = the_Edges_of G1 by A1, GLIB_010:def 11;
    then G1.edgesBetween(F_V"the_Vertices_of H2) c= dom F_E;
    then (SG2SGFunc F).H1 = H2 by A1, A9, GLIB_009:44, GLIBPRE1:101;
    then x = f.y by A7, FUNCT_1:49;
    hence x in rng f by A8, FUNCT_1:def 3;
  end;
  hence thesis by TARSKI:def 3;
end;
