
theorem Th73:
  for G being _finite _Graph holds MCS:CSeq(G)``1 is vertex-numbering
proof
  let G be _finite _Graph;
  set GS = MCS:CSeq(G);
  set S = GS``1;
A1: GS.0 = MCS:Init(G) by Def25;
  S.0 = (GS.0)`1 by Def24;
  hence S.0 = {} by A1;
  now
    let k, n be Nat such that
A2: S.k = S.n;
A3: S.(k+1) = (GS.(k+1))`1 by Def24;
A4: S.k = (GS.k)`1 by Def24;
A5: S.(n+1) = (GS.(n+1))`1 by Def24;
A6: S.n = (GS.n)`1 by Def24;
    per cases;
    suppose
A7:   k <= G.order() & n <= G.order();
      then card dom ((GS.n)`1) = n by Th65;
      hence S.(k+1) = S.(n+1) by A2,A4,A6,A7,Th65;
    end;
    suppose
A8:   k <= G.order() & n >= G.order();
      then
A9:   GS.n = GS.(G.order()) by Th66;
A10:  card dom ((GS.(G.order()))`1) = G.order() by Th65;
A11:  n+1 >= G.order() by A8,NAT_1:13;
      card dom ((GS.k)`1) = k by A8,Th65;
      then k+1 >= G.order() by A2,A4,A6,A9,A10,NAT_1:13;
      hence S.(k+1) = (GS.(G.order()))`1 by A3,Th66
        .= S.(n+1) by A5,A11,Th66;
    end;
    suppose
A12:  k >= G.order() & n <= G.order();
      then
A13:  GS.k = GS.(G.order()) by Th66;
A14:  card dom ((GS.(G.order()))`1) = G.order() by Th65;
      card dom ((GS.n)`1) = n by A12,Th65;
      then
A15:  n+1 >= G.order() by A2,A4,A6,A13,A14,NAT_1:13;
      k+1 >= G.order() by A12,NAT_1:13;
      hence S.(k+1) = (GS.(G.order()))`1 by A3,Th66
        .= S.(n+1) by A5,A15,Th66;
    end;
    suppose
A16:  k >= G.order() & n >= G.order();
      then
A17:  n+1 >= G.order() by NAT_1:13;
A18:  k+1 >= G.order() by A16,NAT_1:13;
      thus S.(k+1) = (GS.(k+1))`1 by Def24
        .= (GS.(G.order()))`1 by A18,Th66
        .= (GS.(n+1))`1 by A17,Th66
        .= S.(n+1) by Def24;
    end;
  end;
  hence S is iterative;
  S is eventually-constant by Th71;
  hence S is halting;
  GS.Lifespan() = S.Lifespan() by Th72;
  hence
A19: S.Lifespan() = G.order() by Th70;
  let n be Nat such that
A20: n < S.Lifespan();
  take w = MCS:PickUnnumbered(GS.n);
A21: (GS.n)`1 = S.n by Def24;
  then dom (S.n) <> the_Vertices_of G by A19,A20,Th69;
  hence not w in dom (S.n) by A21,Th59;
A22: (GS.(n+1))`1 = S.(n+1) by Def24;
  n = card dom (S.n) by A19,A20,A21,Th65;
  hence thesis by A19,A20,A21,A22,Th64;
end;
