
theorem Th62: :: Complete01
  for G being complete _Graph, S being Subset of the_Vertices_of G
  for H being inducedSubgraph of G,S holds H is complete
proof
  let G be complete _Graph;
  let S be Subset of the_Vertices_of G;
  let H be inducedSubgraph of G,S;
  per cases;
  suppose
    S = {};
    then H == G by GLIB_000:def 37;
    hence thesis by Th61;
  end;
  suppose
    S <> {};
    then
A1: the_Vertices_of H = S by GLIB_000:def 37;
    now
      let u,v be Vertex of H such that
A2:   u <> v;
      reconsider v2=v as Vertex of G by A1,TARSKI:def 3;
      reconsider u2=u as Vertex of G by A1,TARSKI:def 3;
      u2,v2 are_adjacent by A2,Def6;
      then consider e being object such that
A3:   e Joins u2,v2,G;
      e Joins u,v,H by A1,A3,Th19;
      hence u,v are_adjacent;
    end;
    hence thesis;
  end;
end;
