
theorem Th82: :: VertexSep01
:: Property "symmetry" for 2 argument modes could be used if we had it
:: as VertexSeparator of a,b is a VertexSeparator of b,a
  for G being _Graph for a,b being Vertex of G st a<>b & not a,b
  are_adjacent for S being VertexSeparator of a,b st S is minimal for H being
removeVertices of G,S for aa being Vertex of H st aa=b for x being Vertex of G
st x in S ex y being Vertex of G st y in H.reachableFrom(aa) & x,y are_adjacent
proof
  let G be _Graph, a,b be Vertex of G such that
A1: a<>b and
A2: not a,b are_adjacent;
  let S be VertexSeparator of a,b such that
A3: S is minimal;
  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), aa be Vertex of H such that
A5: aa=b;
  let x be Vertex of G;
  assume x in S;
  hence thesis by A1,A2,A5,A4,Th81;
end;
