let G be _Graph; :: thesis: 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
for a2 being Vertex of G2 holds (G2 .reachableFrom a2) /\ S = {}
let a, b be Vertex of G; :: thesis: ( a <> b & not a,b are_adjacent implies for S being VertexSeparator of a,b
for G2 being removeVertices of G,S
for a2 being Vertex of G2 holds (G2 .reachableFrom a2) /\ S = {} )
assume that
A1: a <> b and
A2: not a,b are_adjacent ; :: thesis: for S being VertexSeparator of a,b
for G2 being removeVertices of G,S
for a2 being Vertex of G2 holds (G2 .reachableFrom a2) /\ S = {}
let S be VertexSeparator of a,b; :: thesis: for G2 being removeVertices of G,S
for a2 being Vertex of G2 holds (G2 .reachableFrom a2) /\ S = {}
let G2 be removeVertices of G,S; :: thesis: for a2 being Vertex of G2 holds (G2 .reachableFrom a2) /\ S = {}
let a2 be Vertex of G2; :: thesis: (G2 .reachableFrom a2) /\ S = {}
set A = G2 .reachableFrom a2;
not a in S by A1, A2, Def8;
then a in (the_Vertices_of G) \ S by XBOOLE_0:def 5;
then A3: the_Vertices_of G2 = (the_Vertices_of G) \ S by GLIB_000:def 37;

now :: thesis: for x being object holds not x in (G2 .reachableFrom a2) /\ S

let x be object ; :: thesis: not x in (G2 .reachableFrom a2) /\ S
assume A4: x in (G2 .reachableFrom a2) /\ S ; :: thesis: contradiction
A5: x in S by A4, XBOOLE_0:def 4;
x in G2 .reachableFrom a2 by A4, XBOOLE_0:def 4;
hence contradiction by A3, A5, XBOOLE_0:def 5; :: thesis: verum

end;

hence (G2 .reachableFrom a2) /\ S = {} by XBOOLE_0:def 1; :: thesis: verum