set S = (the_Vertices_of G) \ {a,b};
A2: for e being set holds not e Joins a,b,G by A1, Def3;
( a in {a,b} & b in {a,b} ) by TARSKI:def 2;
then A3: ( not a in (the_Vertices_of G) \ {a,b} & not b in (the_Vertices_of G) \ {a,b} ) by XBOOLE_0:def 5;
for G2 being removeVertices of G,((the_Vertices_of G) \ {a,b})
for W being Walk of G2 holds not W is_Walk_from a,b

proof

let G2 be removeVertices of G,((the_Vertices_of G) \ {a,b}); :: thesis: for W being Walk of G2 holds not W is_Walk_from a,b
let W be Walk of G2; :: thesis: not W is_Walk_from a,b
assume A4: W is_Walk_from a,b ; :: thesis: contradiction
not (the_Vertices_of G) \ ((the_Vertices_of G) \ {a,b}) is empty by A3, XBOOLE_0:def 5;
then the_Vertices_of G2 = (the_Vertices_of G) \ ((the_Vertices_of G) \ {a,b}) by GLIB_000:def 39;
then the_Vertices_of G2 = (the_Vertices_of G) /\ {a,b} by XBOOLE_1:48;
then A5: the_Vertices_of G2 = {a,b} by XBOOLE_1:28;

now

let x be set ; :: thesis: ( x in {a,b} implies x in W .vertices() )
assume A6: x in {a,b} ; :: thesis: x in W .vertices()

now

per cases ( x = a or x = b ) by A6, TARSKI:def 2;

suppose x = a ; :: thesis: x in W .vertices()

then x = W .first() by A4, GLIB_001:def 23;
hence x in W .vertices() by GLIB_001:89; :: thesis: verum

end;

suppose x = b ; :: thesis: x in W .vertices()

then x = W .last() by A4, GLIB_001:def 23;
hence x in W .vertices() by GLIB_001:89; :: thesis: verum

end;

end;

end;

hence x in W .vertices() ; :: thesis: verum

end;

then for x being set holds
( x in W .vertices() iff x in {a,b} ) by A5;
then A7: W .vertices() = {a,b} by TARSKI:2;
reconsider W2 = W as Walk of G by GLIB_001:168;

now

let x be set ; :: thesis: ( ( x in W2 .vertices() implies x in W .vertices() ) & ( x in W .vertices() implies x in W2 .vertices() ) )

hereby :: thesis: ( x in W .vertices() implies x in W2 .vertices() )

assume x in W2 .vertices() ; :: thesis: x in W .vertices()
then ex n being odd Element of NAT st
( n <= len W & W . n = x ) by GLIB_001:88;
hence x in W .vertices() by GLIB_001:88; :: thesis: verum

end;

assume x in W .vertices() ; :: thesis: x in W2 .vertices()
then ex n being odd Element of NAT st
( n <= len W2 & W2 . n = x ) by GLIB_001:88;
hence x in W2 .vertices() by GLIB_001:88; :: thesis: verum

end;

then W2 .vertices() = {a,b} by A7, TARSKI:2;
hence contradiction by A1, A2, Th22; :: thesis: verum

end;

hence ex b₁ being Subset of (the_Vertices_of G) st
( not a in b₁ & not b in b₁ & ( for G2 being removeVertices of G,b₁
for W being Walk of G2 holds not W is_Walk_from a,b ) ) by A3; :: thesis: verum