
theorem
  for G being _finite _Graph, n being Nat holds ((MCS:CSeq(G)).n)`1 is
  with_property_T
proof
  let G be _finite _Graph, n be Nat;
  set CN = (MCS:CSeq(G)).n;
  set VLN = CN`1;
  set VL = (MCS:CSeq(G))``1;
  now
A1: (MCS:CSeq(G)).Lifespan() = VL.Lifespan() by Th72;
A2: VLN = VL.n by Def24;
    let a,b,c be Vertex of G such that
A3: a in dom VLN and
A4: b in dom VLN and
A5: c in dom VLN and
A6: VLN.a < VLN.b and
A7: VLN.b < VLN.c and
A8: a,c are_adjacent and
A9: not b,c are_adjacent;
A10: G.order() = (MCS:CSeq(G)).Lifespan() by Th70;
    now
      set bn = G.order() -' VLN.b;
      set CSB = (MCS:CSeq(G)).bn;
      set VLB = CSB`1;
      set VL2 = CSB`2;
      not c in G.AdjacentSet({b}) by A9,CHORD:52;
      then
A11:  not c in (G.AdjacentSet({b}) /\ dom VLB) by XBOOLE_0:def 4;
A12:  b = (MCS:CSeq(G)``1).PickedAt(bn) by A4,A10,A2,A1,Th20;
A13:  c in G.AdjacentSet({a}) by A6,A7,A8,CHORD:52;
A14:  VLB = VL.bn by Def24;
      then not a in dom VLB by A3,A6,A10,A2,A1,Th24;
      then
A15:  VL2.a = card (G.AdjacentSet({a}) /\ dom VLB) by Th76;
      bn < n by A4,A10,A2,A1,Th22;
      then VLB c= VLN by A2,A14,Th17;
      then
A16:  dom VLB c= dom VLN by RELAT_1:11;
      VLN.b <= G.order() by A10,A2,A1,Th15;
      then
A17:  G.order() -' VLN.b = G.order() - VLN.b by XREAL_1:233;
      then VLN.b = G.order() - (G.order() -' VLN.b);
      then
A18:  VLN.b = G.order() -' (G.order() -' VLN.b) by NAT_D:35,XREAL_1:233;
A19:  now
        assume
A20:    a in dom VLB;
        then VLN.b < VLB.a by A10,A1,A14,A18,Th22;
        hence contradiction by A3,A6,A2,A14,A20,Th19;
      end;
A21:  1 <= VLN.b by A4,A2,Th15;
      then
A22:  bn < G.order() by A17,XREAL_1:44;
      then
A23:  dom VLB <> the_Vertices_of G by Th69;
      assume
A24:  for d being Vertex of G st d in dom VLN & VLN.b < VLN.d & b,d
      are_adjacent holds a,d are_adjacent;
      now
        set CSB1 = (MCS:CSeq(G)).(bn+1);
        set VLB1 = CSB1`1;
        let x be object such that
A25:    x in G.AdjacentSet({b}) /\ dom VLB;
        reconsider d = x as Vertex of G by A25;
A26:    x in dom VLB by A25,XBOOLE_0:def 4;
        x in dom VLB by A25,XBOOLE_0:def 4;
        then
A27:    VLN.d = VLB.d by A2,A14,A16,Th19;
A28:    VLB1 = VL.(bn+1) by Def24;
        then b in dom VLB1 by A10,A1,A22,A12,Th11;
        then
A29:    VLN.b = VLB1.b by A4,A2,A28,Th19;
        bn < bn+1 by XREAL_1:39;
        then VLB c= VLB1 by A14,A28,Th17;
        then dom VLB c= dom VLB1 by RELAT_1:11;
        then
A30:    VLB.d = VLB1.d by A14,A26,A28,Th19;
        VLB.d in rng VLB by A26,FUNCT_1:def 3;
        then VLB.d in (Seg G.order()\Seg (G.order()-'bn)) by A10,A1,A14,Th14;
        then G.order() -' bn < VLB1.d by A30,Th3;
        then
A31:    VLN.b < VLN.d by A10,A1,A17,A21,A12,A28,A29,A27,A30,Th12,XREAL_1:44;
        d in G.AdjacentSet({b}) by A25,XBOOLE_0:def 4;
        then b,d are_adjacent by CHORD:52;
        then a,d are_adjacent by A24,A16,A26,A31;
        then d in G.AdjacentSet({a}) by A6,A31,CHORD:52;
        hence x in G.AdjacentSet({a}) /\ dom VLB by A26,XBOOLE_0:def 4;
      end;
      then
A32:  (G.AdjacentSet({b}) /\ dom VLB) c= (G.AdjacentSet({a}) /\ dom VLB);
      c in dom VLB by A4,A5,A7,A10,A2,A1,A14,Th23;
      then c in (G.AdjacentSet({a}) /\ dom VLB) by A13,XBOOLE_0:def 4;
      then
A33:  (G.AdjacentSet({b}) /\ dom VLB) c< (G.AdjacentSet({a}) /\ dom VLB)
      by A11,A32,XBOOLE_0:def 8;
A34:  b = MCS:PickUnnumbered(CSB) by A17,A21,A12,Th74,XREAL_1:44;
      then VL2.b = card (G.AdjacentSet({b}) /\ dom VLB) by A23,Th59,Th76;
      hence contradiction by A23,A34,A19,A15,A33,Th58,TREES_1:6;
    end;
    hence ex d being Vertex of G st d in dom VLN & VLN.b < VLN.d & b,d
    are_adjacent & not a,d are_adjacent;
  end;
  hence thesis;
end;
