
theorem Th66: :: Simplicial2
  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 u is
  simplicial iff v is simplicial
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;
A4: G.AdjacentSet({u}) = {} iff G2.AdjacentSet({v}) = {} by A1,A2,A3,Th57;
  per cases;
  suppose
    G.AdjacentSet({u}) = {};
    hence thesis by A4;
  end;
  suppose
A5: G.AdjacentSet({u}) <> {};
    hereby
      set Ga = the AdjGraph of G,{u};
      assume u is simplicial;
      then
A6:   Ga is complete by A5;
      thus v is simplicial
      by A1,A2,A3,A5,A6,Th59,Th61;
    end;
    set Ha = the AdjGraph of G2,{v};
    assume
A7: v is simplicial;
    G2.AdjacentSet({v}) <> {} by A1,A2,A3,A5,Th57;
    then
A8: Ha is complete by A7;
    thus u is simplicial
    proof
      assume G.AdjacentSet({u}) <> {};
      let Ga be AdjGraph of G,{u};
      Ga == Ha by A1,A2,A3,A5,Th59;
      hence thesis by A8,Th61;
    end;
  end;
end;
