
theorem
  for G2, H being _Graph, F being PGraphMapping of G2, H
  st F is weak_SG-embedding ex G1 being Supergraph of G2 st G1 is H-isomorphic
proof
  let G2, H be _Graph, F be PGraphMapping of G2, H;
  assume A1: F is weak_SG-embedding;
  then reconsider F as one-to-one PGraphMapping of G2, H;
  F"_E is one-to-one;
  then consider E being Subset of the_Edges_of G2 such that
    A2: for G4 being reverseEdgeDirections of G2, E
      ex F9 being PGraphMapping of H, G4 st F9 = F" & F9 is directed &
        (F" is non empty  implies F9 is non empty) &
        (F" is total implies F9 is total) &
        (F" is one-to-one implies F9 is one-to-one) &
        (F" is onto implies F9 is onto) &
        (F" is semi-continuous implies F9 is semi-continuous) &
        (F" is continuous implies F9 is continuous) by GLIBPRE0:86;
  set G4 = the reverseEdgeDirections of G2, E;
  consider F9 being PGraphMapping of H, G4 such that
    A3: F9 = F" & F9 is directed and
    (F" is non empty  implies F9 is non empty) &
    (F" is total implies F9 is total) and
     A4: (F" is one-to-one implies F9 is one-to-one) and
     A5: (F" is onto implies F9 is onto) and
    (F" is semi-continuous implies F9 is semi-continuous) &
    (F" is continuous implies F9 is continuous) by A2;
  reconsider F9 as one-to-one PGraphMapping of H, G4 by A4;
  A6: F9 is onto by A1, A5, GLIB_010:71;
  then F9" is total by GLIB_010:72;
  then A7: F9" is weak_SG-embedding;
  F9 is Dcontinuous by A3, A6;
  then F9" is semi-Dcontinuous;
  then consider G3 being Supergraph of G4 such that
    A8: G3 is H-Disomorphic by A7, Th114;
  set G1 = the reverseEdgeDirections of G3, E;
  A9: G2 is reverseEdgeDirections of G4, E by GLIB_007:3;
  the_Edges_of G4 = the_Edges_of G2 by GLIB_007:4;
  then reconsider G1 as Supergraph of G2 by A9, GLIBPRE0:50;
  take G1;
  G1 is G3-isomorphic & G3 is H-isomorphic by A8, GLIBPRE0:78;
  hence thesis;
end;
