let G2 be _Graph; for v1, e being object
for v2 being Vertex of G2
for G1 being addAdjVertex of G2,v1,e,v2 st not v1 in the_Vertices_of G2 & not e in the_Edges_of G2 holds
v1 is Vertex of G1
let v1, e be object ; for v2 being Vertex of G2
for G1 being addAdjVertex of G2,v1,e,v2 st not v1 in the_Vertices_of G2 & not e in the_Edges_of G2 holds
v1 is Vertex of G1
let v2 be Vertex of G2; for G1 being addAdjVertex of G2,v1,e,v2 st not v1 in the_Vertices_of G2 & not e in the_Edges_of G2 holds
v1 is Vertex of G1
let G1 be addAdjVertex of G2,v1,e,v2; ( not v1 in the_Vertices_of G2 & not e in the_Edges_of G2 implies v1 is Vertex of G1 )
assume
( not v1 in the_Vertices_of G2 & not e in the_Edges_of G2 )
; v1 is Vertex of G1
then A1:
the_Vertices_of G1 = (the_Vertices_of G2) \/ {v1}
by Def14;
v1 in {v1}
by TARSKI:def 1;
hence
v1 is Vertex of G1
by A1, XBOOLE_0:def 3; verum