
theorem Th59: :: Simplicial2b
  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 & G.AdjacentSet({u}) <> {} for v being Vertex
of H st u=v for Ga being AdjGraph of G,{u}, Ha being AdjGraph of H,{v} holds Ga
  == Ha
proof
  let G be _Graph;
  let S being non empty Subset of the_Vertices_of G;
  let H be inducedSubgraph of G,S;
  let u be Vertex of G such that
A1: u in S and
A2: G.AdjacentSet({u}) c= S and
A3: G.AdjacentSet({u}) <> {};
  let v be Vertex of H;
  assume u=v;
  then
A4: G.AdjacentSet({u}) = H.AdjacentSet({v}) by A1,A2,Th57;
  let Ga be AdjGraph of G,{u}, Ha being AdjGraph of H,{v};
A5: Ha is inducedSubgraph of H,H.AdjacentSet({v}) by Def5;
A6: Ga is inducedSubgraph of G,G.AdjacentSet({u}) by Def5;
  then
A7: the_Edges_of Ga = G.edgesBetween(G.AdjacentSet({u})) by A3,GLIB_000:def 37;
  the_Vertices_of Ga = G.AdjacentSet({u}) by A3,A6,GLIB_000:def 37;
  hence the_Vertices_of Ga = the_Vertices_of Ha by A4,A5,GLIB_000:def 37;
  the_Edges_of Ha = H.edgesBetween(H.AdjacentSet({v})) by A3,A4,A5,
GLIB_000:def 37;
  hence
A8: the_Edges_of Ga = the_Edges_of Ha by A2,A3,A4,A7,Th31;
A9: the_Target_of Ga = (the_Target_of G)|the_Edges_of Ga by GLIB_000:45;
  Ha is inducedSubgraph of H,G.AdjacentSet({u}) by A4,Def5;
  then
A10: Ha is inducedSubgraph of G,G.AdjacentSet({u}) by A2,A3,Th29;
  the_Source_of Ga = (the_Source_of G)|the_Edges_of Ga by GLIB_000:45;
  hence the_Source_of Ha = the_Source_of Ga & the_Target_of Ha = the_Target_of
  Ga by A8,A9,A10,GLIB_000:45;
end;
