
theorem Th46:
  for G being _finite _Graph, m,n being Nat st m < n for x being
set st not G.order()-'m in ((LexBFS:CSeq(G)).(m+1))`2.x holds not G.order()-'m
  in ((LexBFS:CSeq(G)).n)`2.x
proof
  let G be _finite _Graph, m,n be Nat;
  assume m < n;
  then m+1 <= n by NAT_1:13;
  then
A1: ex j being Nat st m+1+j = n by NAT_1:10;
  set CS = LexBFS:CSeq(G);
  set CSM = CS.(m+1);
  set V2M = CSM`2;
  let x be set such that
A2: not G.order() -' m in V2M.x;
  defpred P[Nat] means not G.order() -' m in (((LexBFS:CSeq(G)).(m+1+$1))`2).x;
A3: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A4: P[k];
    set CSK = CS.(m+1+k);
    set VLK = CSK`1, V2K = CSK`2, CK1 = CS.(m+1+k+1);
    set V21 = CK1`2;
    now
      per cases;
      suppose
A5:     m+1+k < G.order();
        then consider w being Vertex of G such that
        w = LexBFS:PickUnnumbered(CSK) and
A6:     for v being set holds ( v in G.AdjacentSet({w}) & not v in dom
VLK implies V21.v = V2K.v \/ {G.order() -' (m+1+k)}) & ((not v in G.AdjacentSet
        ({w}) or v in dom VLK) implies V21.v = V2K.v) by Th42;
        per cases;
        suppose
A7:       x in G.AdjacentSet({w}) & not x in dom VLK;
          m+1 <= m+1+k by NAT_1:11;
          then m < m+1+k by XREAL_1:39;
          then G.order() -' m > G.order() -' (m+1+k) by A5,Th2;
          then
A8:       not G.order() -' m in {G.order() -' (m+1+k)} by TARSKI:def 1;
          V21.x = V2K.x \/ {G.order() -' (m+1+k)} by A6,A7;
          hence not G.order() -' m in V21.x by A4,A8,XBOOLE_0:def 3;
        end;
        suppose
          not x in G.AdjacentSet({w}) or x in dom VLK;
          hence not G.order() -' m in V21.x by A4,A6;
        end;
      end;
      suppose
A9:     G.order() <= m+1+k;
        m+1+k <= m+1+k+1 by NAT_1:13;
        hence not G.order() -' m in V21.x by A4,A9,Th34;
      end;
    end;
    hence thesis;
  end;
A10: P[ 0 ] by A2;
  for k being Nat holds P[k] from NAT_1:sch 2(A10,A3);
  hence thesis by A1;
end;
