
theorem Th47:
  for G being _finite _Graph, m,n,k being Nat st k < n & n <= m for
x being set st not G.order()-'k in ((LexBFS:CSeq(G)).n)`2.x holds not G.order()
  -'k in ((LexBFS:CSeq(G)).m)`2.x
proof
  let G be _finite _Graph, m,n,k be Nat such that
A1: k < n and
A2: n <= m;
  set CS = LexBFS:CSeq(G);
  set CSN = CS.n;
  set V2N = CSN`2;
  let x be set such that
A3: not G.order() -' k in V2N.x;
  set CK1 = CS.(k+1), V21 = CK1`2;
  k+1 <= n by A1,NAT_1:13;
  then V21.x c= V2N.x by Th44;
  then
A4: not G.order() -' k in V21.x by A3;
  k < m by A1,A2,XXREAL_0:2;
  hence thesis by A4,Th46;
end;
