
theorem Th65: :: Simplicial1
  for G1,G2 being _Graph st G1 == G2 for u1 being Vertex of G1, u2
  being Vertex of G2 st u1=u2 & u1 is simplicial holds u2 is simplicial
proof
  let G1,G2 be _Graph such that
A1: G1 == G2;
  let u1 be Vertex of G1, u2 be Vertex of G2 such that
A2: u1 = u2;
  assume
A3: u1 is simplicial;
  now
    per cases;
    suppose
      G1.AdjacentSet({u1}) = {};
      then G2.AdjacentSet({u2}) = {} by A1,A2,Th50;
      hence thesis;
    end;
    suppose
A4:   G1.AdjacentSet({u1}) <> {};
      set H1 = the AdjGraph of G1,{u1};
      H1 is complete by A3,A4;
      then for H2 being AdjGraph of G2,{u2} holds H2 is complete by A1,A2,Th58
,Th61;
      hence thesis;
    end;
  end;
  hence thesis;
end;
