let G2 be _Graph; :: thesis: 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 ; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ) ; :: thesis: 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; :: thesis: verum