
theorem Th145:
  for G3, G4 being _Graph, V1, V2 being set
  for G1 being addVertices of G3,V1, G2 being addVertices of G4,V2
  for F0 being PGraphMapping of G3,G4, f being one-to-one Function
  st dom f = V1 \ the_Vertices_of G3 & rng f = V2 \ the_Vertices_of G4
  ex F being PGraphMapping of G1, G2 st F = [F0_V +* f, F0_E] &
    (F0 is weak_SG-embedding implies F is weak_SG-embedding) &
    (F0 is strong_SG-embedding implies F is strong_SG-embedding) &
    (F0 is isomorphism implies F is isomorphism) &
    (F0 is Disomorphism implies F is Disomorphism)
proof
  let G3, G4 be _Graph, V1, V2 be set;
  let G1 be addVertices of G3,V1, G2 be addVertices of G4,V2;
  let F0 be PGraphMapping of G3,G4, f be one-to-one Function;
  assume dom f = V1 \ the_Vertices_of G3 & rng f = V2 \ the_Vertices_of G4;
  then consider F being PGraphMapping of G1, G2 such that
    A1: F = [F0_V +* f, F0_E] and
    F0 is non empty  implies F is non empty and
    A2: F0 is total implies F is total and
    A3: F0 is onto implies F is onto and
    A4: F0 is one-to-one implies F is one-to-one and
    A5: F0 is directed implies F is directed and
    F0 is semi-continuous implies F is semi-continuous and
    A6: F0 is continuous implies F is continuous and
    F0 is semi-Dcontinuous implies F is semi-Dcontinuous and
    F0 is Dcontinuous implies F is Dcontinuous by Th144;
  take F;
  thus F = [F0_V +* f, F0_E] by A1;
  thus F0 is weak_SG-embedding implies F is weak_SG-embedding by A2, A4;
  thus F0 is strong_SG-embedding implies F is strong_SG-embedding
    by A2, A4, A6;
  thus F0 is isomorphism implies F is isomorphism by A2, A3, A4;
  thus F0 is Disomorphism implies F is Disomorphism by A2, A3, A4, A5;
end;
