
theorem Th114:
for G being SimpleGraph, n be Nat holds (MycielskianSeq G).n is SimpleGraph
proof
  let G be SimpleGraph, n be Nat;
  set MG = MycielskianSeq G;
  defpred P[Nat] means MG.$1 is SimpleGraph;
  consider myc being Function such that
A1: MycielskianSeq G = myc and
A2: myc.0 = G and
A3: for k being Nat, G being SimpleGraph
       st G = myc.k holds myc.(k+1) = Mycielskian G by Def26;
A4: P[0] by A1,A2;
A5: for k being Nat st P[k] holds P[k+1] proof
      let k be Nat;
      assume P[k];
       then reconsider H = MG.k as SimpleGraph;
       MG.(k+1) = Mycielskian H by A1,A3;
      hence P[k+1];
    end;
  for k being Nat holds P[k] from NAT_1:sch 2(A4,A5);
  hence thesis;
end;
