
theorem Th48:
  for G being _finite _Graph, m,n being Nat, x being Vertex of G st
n in ((LexBFS:CSeq(G)).m)`2.x ex y being Vertex of G st (LexBFS:PickUnnumbered(
  (LexBFS:CSeq(G)).(G.order()-'n))) = y & not y in dom ((LexBFS:CSeq(G)).(G
  .order()-'n))`1 & x in G.AdjacentSet({y})
proof
  let G be _finite _Graph, m, n be Nat;
  set CS = LexBFS:CSeq(G);
  set CSM = CS.m;
  set V2M = CSM`2;
  set CSN = CS.(G.order() -' n);
  set VLN = CSN`1;
  set V2N = CSN`2;
  set on1 = G.order() -' n + 1;
  set CN1 = CS.on1;
  set V21 = CN1`2;
  let x be Vertex of G such that
A1: n in V2M.x;
A2: V2M.x c= (Seg G.order()) \ Seg (G.order() -' m) by Th43;
  then
A3: G.order() -' m < n by A1,Th3;
  n <= G.order() by A1,A2,Th3;
  then
A4: G.order() -' n = G.order() - n by XREAL_1:233;
  then
A5: G.order() -' n < G.order() by A3,XREAL_1:44;
  then
A6: G.order() -' (G.order() -' n) = G.order() - (G.order() - n) by A4,
XREAL_1:233;
  then consider w being Vertex of G such that
A7: w = LexBFS:PickUnnumbered(CSN) and
A8: for v being set holds (v in G.AdjacentSet({w}) & not v in dom VLN
  implies V21.v = V2N.v\/{n}) & (not v in G.AdjacentSet({w}) or v in dom VLN
  implies V21.v = V2N.v) by A3,A4,Th42,XREAL_1:44;
  V2N.x c= (Seg G.order()) \ Seg (G.order() -' (G.order() -' n)) by Th43;
  then
A9: not n in V2N.x by A6,Th3;
A10: now
    per cases;
    suppose
      m <= G.order();
      then G.order() -' m = G.order() - m by XREAL_1:233;
      then G.order() - m + m < n + m by A3,XREAL_1:6;
      then G.order() - n < m + n - n by XREAL_1:9;
      hence on1 <= m by A4,NAT_1:13;
    end;
    suppose
      G.order() < m;
      then G.order() -' n < m by A5,XXREAL_0:2;
      hence on1 <= m by NAT_1:13;
    end;
  end;
A11: G.order() -' n < on1 by XREAL_1:39;
  assume
A12: not ex y being Vertex of G st LexBFS:PickUnnumbered(CSN) = y & not y
  in dom VLN & x in G.AdjacentSet({y});
  dom VLN <> the_Vertices_of G by A5,Th36;
  then not x in G.AdjacentSet({w}) by A12,A7,Th30;
  then not n in V21.x by A9,A8;
  hence contradiction by A1,A6,A10,A11,Th47;
end;
