
theorem
  for G3, G4 being _Graph, v1, v2 being object
  for G1 being addVertex of G3,v1, G2 being addVertex of G4,v2
  for F0 being PGraphMapping of G3,G4
  st not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4
  ex F being PGraphMapping of G1, G2 st F = [F0_V +* (v1 .--> v2), 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 object;
  let G1 be addVertex of G3,v1, G2 be addVertex of G4,v2;
  let F0 be PGraphMapping of G3,G4;
  assume not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4;
  then consider F being PGraphMapping of G1, G2 such that
    A1: F = [F0_V +* (v1 .--> v2), F0_E] 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 Th148;
  take F;
  thus F = [F0_V +* (v1 .--> v2), 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;
