let G be _Graph; for a, b being Vertex of G st a <> b & not a,b are_adjacent holds
for S being VertexSeparator of a,b
for G2 being removeVertices of G,S holds
( a is Vertex of G2 & b is Vertex of G2 )
let a, b be Vertex of G; ( a <> b & not a,b are_adjacent implies for S being VertexSeparator of a,b
for G2 being removeVertices of G,S holds
( a is Vertex of G2 & b is Vertex of G2 ) )
assume that
A1:
a <> b
and
A2:
not a,b are_adjacent
; for S being VertexSeparator of a,b
for G2 being removeVertices of G,S holds
( a is Vertex of G2 & b is Vertex of G2 )
let S be VertexSeparator of a,b; for G2 being removeVertices of G,S holds
( a is Vertex of G2 & b is Vertex of G2 )
let G2 be removeVertices of G,S; ( a is Vertex of G2 & b is Vertex of G2 )
not b in S
by A1, A2, Def8;
then A3:
b in (the_Vertices_of G) \ S
by XBOOLE_0:def 5;
not a in S
by A1, A2, Def8;
then
a in (the_Vertices_of G) \ S
by XBOOLE_0:def 5;
hence
( a is Vertex of G2 & b is Vertex of G2 )
by A3, GLIB_000:def 37; verum