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

theorem Th132:
  for F being PGraphMapping of G1, G2 st F is total
  holds rng(SG2SGFunc(F) | G1.allConnectedSG()) c= G2.allConnectedSG()
proof
  let F be PGraphMapping of G1, G2;
  assume A1: F is total;
  set f = SG2SGFunc(F) | G1.allConnectedSG();
  now
    let y be object;
    assume A2: y in rng f;
    then consider x being object such that
      A3: x in dom f & f.x = y by FUNCT_1:def 3;
    reconsider H1 = x as plain connected Subgraph of G1 by A3, Th124;
    y in rng SG2SGFunc(F) by A2, TARSKI:def 3, RELAT_1:70;
    then reconsider H2 = y as plain Subgraph of G2 by Th1;
    A4: H2 = (SG2SGFunc F).x by A3, FUNCT_1:47
      .= rng(F|H1) by Def5;
    now
      let v2,w2 be Vertex of H2;
      A5: F | H1 is total by A1, GLIB_010:57;
      A6: the_Vertices_of H2 = rng(F|H1)_V by A4, A5, GLIB_010:54;
      then consider v1 being object such that
        A7: v1 in dom(F|H1)_V & ((F|H1)_V).v1 = v2 by FUNCT_1:def 3;
      consider w1 being object such that
        A8: w1 in dom(F|H1)_V & ((F|H1)_V).w1 = w2 by A6, FUNCT_1:def 3;
      reconsider v1, w1 as Vertex of H1 by A7, A8;
      consider W1 being Walk of H1 such that
        A9: W1 is_Walk_from v1,w1 by GLIB_002:def 1;
      reconsider F9 = F | H1 as non empty PGraphMapping of H1, rng(F|H1)
        by A5, GLIBPRE1:88;
      F9 is total by A1, GLIBPRE1:109;
      then reconsider W1 as F9-defined Walk of H1 by GLIB_010:121;
      reconsider W2 = F9.:W1 as Walk of H2 by A4;
      take W2;
      F9.:W1 is_Walk_from F9_V.v1, F9_V.w1 by A9, GLIB_010:132;
      hence W2 is_Walk_from v2, w2 by A7, A8, GLIB_001:19;
    end;
    then H2 is connected by GLIB_002:def 1;
    hence y in G2.allConnectedSG() by Th124;
  end;
  hence thesis by TARSKI:def 3;
end;
