
theorem Th57: :: Simplicial2a
  for G being _Graph for S being non empty Subset of
the_Vertices_of G for H being inducedSubgraph of G,S for u being Vertex of G st
  u in S & G.AdjacentSet({u}) c= S for v being Vertex of H st u=v holds G
  .AdjacentSet({u}) = H.AdjacentSet({v})
proof
  let G be _Graph;
  let S be non empty Subset of the_Vertices_of G;
  let G2 be inducedSubgraph of G,S;
  let u be Vertex of G such that
A1: u in S and
A2: G.AdjacentSet({u}) c= S;
  let v be Vertex of G2 such that
A3: u=v;
  now
    let x be object;
    hereby
      assume
A4:   x in G.AdjacentSet({u});
      then reconsider y=x as Vertex of G;
      reconsider w=x as Vertex of G2 by A2,A4,GLIB_000:def 37;
      y,u are_adjacent by A4,Th51;
      then consider e being object such that
A5:   e Joins y,u,G;
      e Joins y,u,G2 by A1,A2,A4,A5,Th19;
      then
A6:   w,v are_adjacent by A3;
      w <> v by A3,A4,Th51;
      hence x in G2.AdjacentSet({v}) by A6,Th51;
    end;
    assume
A7: x in G2.AdjacentSet({v});
    then reconsider y=x as Vertex of G2;
    x in the_Vertices_of G2 by A7;
    then reconsider w=x as Vertex of G;
    y,v are_adjacent by A7,Th51;
    then consider e being object such that
A8: e Joins y,v,G2;
    e Joins y,v,G by A8,GLIB_000:72;
    then
A9: w,u are_adjacent by A3;
    w <> u by A3,A7,Th51;
    hence x in G.AdjacentSet({u}) by A9,Th51;
  end;
  hence thesis by TARSKI:2;
end;
