let G3 be _Graph; :: thesis:
let G4 be G3 -isomorphic _Graph; :: thesis:
let v1, v2 be object ; :: thesis:
let G1 be addAdjVertexAll of G3,v1; :: thesis:
let G2 be addAdjVertexAll of G4,v2; :: thesis:
assume ( v1 in the_Vertices_of G3 iff v2 in the_Vertices_of G4 ) ; :: thesis:

suppose A1: ( not v1 in the_Vertices_of G3 & not v2 in the_Vertices_of G4 ) ; :: thesis:

consider F0 being PGraphMapping of G3,G4 such that
A2: F0 is isomorphism by Def23;
A3: (F0 _V) | (the_Vertices_of G3) is one-to-one by A2;
A4: dom ((F0 _V) | (the_Vertices_of G3)) = the_Vertices_of G3 by A2, Def11;
rng ((F0 _V) | (the_Vertices_of G3)) = the_Vertices_of G4 by A2, Def12;
then consider F being PGraphMapping of G1,G2 such that
( F _V = (F0 _V) +* (v1 .--> v2) & (F _E) | (dom (F0 _E)) = F0 _E ) and
( F0 is total implies F is total ) and
( F0 is onto implies F is onto ) and
( F0 is one-to-one implies F is one-to-one ) and
( F0 is weak_SG-embedding implies F is weak_SG-embedding ) and
A5: ( F0 is isomorphism implies F is isomorphism ) by A1, A3, A4, Th162;
thus G2 is G1 -isomorphic by A2, A5; :: thesis:

end;

suppose ( v1 in the_Vertices_of G3 & v2 in the_Vertices_of G4 ) ; :: thesis:

then ( G1 == G3 & G2 == G4 ) by GLIB_007:def 4;
then ( G1 is reverseEdgeDirections of G3, {} & G2 is reverseEdgeDirections of G4, {} ) by GLIB_009:42;
hence G2 is G1 -isomorphic by Th143; :: thesis:

end;

end;