
theorem
  for G3, G4 being _Graph, v3 being Vertex of G3, v4 being Vertex of G4
  for e1,e2,v1,v2 being object
  for G1 being addAdjVertex of G3,v1,e1,v3
  for G2 being addAdjVertex of G4,v2,e2,v4
  for F0 being PGraphMapping of G3, G4
  st not e1 in the_Edges_of G3 & not e2 in the_Edges_of G4 &
    not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 &
    v3 in dom F0_V & F0_V.v3 = v4
  ex F being PGraphMapping of G1, G2 st
    F = [F0_V +* (v1 .--> v2), F0_E +* (e1 .--> e2)] &
    (F0 is weak_SG-embedding implies F is weak_SG-embedding) &
    (F0 is isomorphism implies F is isomorphism) &
    (F0 is Disomorphism implies F is Disomorphism)
proof
  let G3, G4 be _Graph, v3 be Vertex of G3, v4 be Vertex of G4;
  let e1,e2,v1,v2 be object;
  let G1 be addAdjVertex of G3,v1,e1,v3, G2 be addAdjVertex of G4,v2,e2,v4;
  let F0 be PGraphMapping of G3, G4;
  assume not e1 in the_Edges_of G3 & not e2 in the_Edges_of G4 &
    not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 &
    v3 in dom F0_V & F0_V.v3 = v4;
  then consider F being PGraphMapping of G1, G2 such that
    A1: F = [F0_V +* (v1 .--> v2), F0_E +* (e1 .--> e2)] 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 by Th155;
  take F;
  thus F = [F0_V +* (v1 .--> v2), F0_E +* (e1 .--> e2)] by A1;
  thus F0 is weak_SG-embedding implies F is weak_SG-embedding by A2, A4;
  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;
