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

theorem Th51:
  for F being PGraphMapping of G1, G2 st F is total continuous
  holds rng(SG2SGFunc(F) | G1.allInducedSG()) c= G2.allInducedSG()
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is total continuous;
  set f = SG2SGFunc(F) | G1.allInducedSG();
  A2: dom f = G1.allInducedSG() by FUNCT_2:def 1;
  now
    let x be object;
    assume x in rng f;
    then consider y being object such that
      A3: y in dom f & x = f.y by FUNCT_1:def 3;
    consider V1 being non empty Subset of the_Vertices_of G1 such that
      A4: y = the plain inducedSubgraph of G1, V1 by A2, A3;
    reconsider H1 = y as inducedSubgraph of G1, V1 by A4;
    dom F_V = the_Vertices_of G1 by A1, GLIB_010:def 11;
    then A5: F_V.:V1 is non empty Subset of the_Vertices_of G2 by RELAT_1:119;
    G1 == dom F by A1, GLIB_010:55;
    then A6: the_Vertices_of G1 = the_Vertices_of dom F by GLIB_000:def 34;
    x = SG2SGFunc(F).y by A3, FUNCT_1:47
      .= rng(F | H1) by A4, Def5;
    then x is plain inducedSubgraph of G2, F_V.:V1 by A1, A6, GLIBPRE1:102;
    hence x in G2.allInducedSG() by A5, Th45;
  end;
  hence thesis by TARSKI:def 3;
end;
