
theorem Th45:
  for G being _finite _Graph, m,n being Nat, x, y being set st n <
  G.order() & n < m & y = LexBFS:PickUnnumbered((LexBFS:CSeq(G)).n) & not x in
dom ((LexBFS:CSeq(G)).n)`1 & x in G.AdjacentSet({y}) holds (G.order() -' n) in
  ((LexBFS:CSeq(G)).m)`2.x
proof
  let G be _finite _Graph;
  let m,n be Nat, x, y be set such that
A1: n < G.order() and
A2: n < m;
  set CS = (LexBFS:CSeq(G));
  set CSM = CS.m, V2M = CSM`2;
  set CN1 = CS.(n+1);
  set V21 = CN1`2;
  n+1 <= m by A2,NAT_1:13;
  then
A3: V21.x c= V2M.x by Th44;
A4: G.order() -' n in {G.order() -' n} by TARSKI:def 1;
  set CSN = CS.n, VLN = CSN`1, V2N = CSN`2;
  assume that
A5: y = LexBFS:PickUnnumbered(CSN) and
A6: not x in dom VLN and
A7: x in G.AdjacentSet({y});
  ex w being Vertex of G st w = LexBFS:PickUnnumbered(CSN) & for v
being set holds (v in G.AdjacentSet({w}) & not v in dom VLN implies V21.v = V2N
.v \/ {G.order() -' n}) & (not v in G.AdjacentSet({w}) or v in dom VLN implies
  V21.v = V2N.v) by A1,Th42;
  then V21.x = V2N.x \/ {G.order() -' n} by A5,A6,A7;
  then G.order() -' n in V21.x by A4,XBOOLE_0:def 3;
  hence thesis by A3;
end;
