
theorem Th66:
  for G being _finite _Graph, n being Nat st G.order() <= n holds (
  MCS:CSeq(G)).(G.order()) = (MCS:CSeq(G)).n
proof
  let G be _finite _Graph, n be Nat;
  assume G.order() <= n;
  then
A1: ex i being Nat st G.order() + i = n by NAT_1:10;
  set CS = MCS:CSeq(G);
  defpred V[Nat] means G.order() = card (dom (CS.(G.order()+$1))`1);
  defpred P[Nat] means (CS.(G.order())) = (CS.(G.order()+$1));
A2: for k being Nat st V[k] holds V[k+1]
  proof
    let k be Nat such that
A3: V[k];
    set CK1 = (MCS:CSeq(G)).(G.order()+k+1);
    set CSK = (MCS:CSeq(G)).(G.order()+k);
    CK1 = MCS:Step(CSK) by Def25;
    hence thesis by A3,Def22;
  end;
A4: V[ 0 ] by Th65;
A5: for k being Nat holds V[k] from NAT_1:sch 2(A4,A2);
A6: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A7: P[k];
    set CK1 = (MCS:CSeq(G)).(G.order()+k+1);
    set CSK = (MCS:CSeq(G)).(G.order()+k);
    set VLK = CSK`1;
A8: CK1 = MCS:Step(CSK) by Def25;
    card dom VLK = G.order() by A5;
    hence thesis by A7,A8,Def22;
  end;
A9: P[ 0 ];
  for k being Nat holds P[k] from NAT_1:sch 2(A9,A6);
  hence thesis by A1;
end;
