
theorem Th67: :: Simplicial03
  for G being _Graph, v being Vertex of G st v is simplicial
for a,b being object
   st a<>b & a in G.AdjacentSet({v}) & b in G.AdjacentSet({v}) holds
  ex e being object st e Joins a,b,G
proof
  let G be _Graph, x be Vertex of G such that
A1: x is simplicial;
  set H = the AdjGraph of G,{x};
  let a,b be object such that
A2: a<>b and
A3: a in G.AdjacentSet({x}) and
A4: b in G.AdjacentSet({x});
A5: H is inducedSubgraph of G,G.AdjacentSet({x}) by Def5;
  then reconsider u=a as Vertex of H by A3,GLIB_000:def 37;
  reconsider v=b as Vertex of H by A4,A5,GLIB_000:def 37;
  H is complete by A1,A3;
  then u,v are_adjacent by A2;
  then consider e being object such that
A6: e Joins u,v,H;
  e Joins a,b,G by A6,GLIB_000:72;
  hence thesis;
end;
