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 st S is minimal holds

for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let a, b be Vertex of G; :: thesis: ( a <> b & not a,b are_adjacent implies for S being VertexSeparator of a,b st S is minimal holds

for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume that

A1: a <> b and

A2: not a,b are_adjacent ; :: thesis: for S being VertexSeparator of a,b st S is minimal holds

for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let S be VertexSeparator of a,b; :: thesis: ( S is minimal implies for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume A3: S is minimal ; :: thesis: for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

reconsider S1 = S as VertexSeparator of b,a by A1, A2, Th69;

A4: S1 is minimal by A1, A2, A3, Th79;

let H be removeVertices of G,S; :: thesis: for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let aa be Vertex of H; :: thesis: ( aa = b implies for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume A5: aa = b ; :: thesis: for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let x be Vertex of G; :: thesis: ( x in S implies ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume x in S ; :: thesis: ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

hence ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) by A1, A2, A5, A4, Th81; :: thesis: verum

for S being VertexSeparator of a,b st S is minimal holds

for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let a, b be Vertex of G; :: thesis: ( a <> b & not a,b are_adjacent implies for S being VertexSeparator of a,b st S is minimal holds

for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume that

A1: a <> b and

A2: not a,b are_adjacent ; :: thesis: for S being VertexSeparator of a,b st S is minimal holds

for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let S be VertexSeparator of a,b; :: thesis: ( S is minimal implies for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume A3: S is minimal ; :: thesis: for H being removeVertices of G,S

for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

reconsider S1 = S as VertexSeparator of b,a by A1, A2, Th69;

A4: S1 is minimal by A1, A2, A3, Th79;

let H be removeVertices of G,S; :: thesis: for aa being Vertex of H st aa = b holds

for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let aa be Vertex of H; :: thesis: ( aa = b implies for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume A5: aa = b ; :: thesis: for x being Vertex of G st x in S holds

ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

let x be Vertex of G; :: thesis: ( x in S implies ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) )

assume x in S ; :: thesis: ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent )

hence ex y being Vertex of G st

( y in H .reachableFrom aa & x,y are_adjacent ) by A1, A2, A5, A4, Th81; :: thesis: verum