let G1, G2 be _Graph; :: thesis:
let v be set ; :: thesis:
assume that
A1: G1 == G2 and
A2: ( G1 is _trivial or not v in the_Vertices_of G1 ) ; :: thesis:
A3: G2 is Subgraph of G1 by A1, GLIB_000:87;
set V = (the_Vertices_of G1) \ {v};

per cases by A2;

suppose G1 is _trivial ; :: thesis:

then consider v1 being Vertex of G1 such that
A4: the_Vertices_of G1 = {v1} by GLIB_000:22;

suppose v = v1 ; :: thesis:

then (the_Vertices_of G1) \ {v} = {} by A4, XBOOLE_1:37;
hence G2 is removeVertex of G1,v by A1, A3, GLIB_000:def 37; :: thesis:

end;

suppose v <> v1 ; :: thesis:

then not v in the_Vertices_of G1 by A4, TARSKI:def 1;
hence G2 is removeVertex of G1,v by A1, Lm3; :: thesis:

end;

end;

end;

suppose not v in the_Vertices_of G1 ; :: thesis:

hence G2 is removeVertex of G1,v by A1, Lm3; :: thesis:

end;

end;