
theorem Th27:
  for G being _finite _Graph, L being LexBFS:Labeling of G, v be
Vertex of G, x being set, k being Nat st x in G.AdjacentSet({v}) & not x in dom
  L`1 holds LexBFS:Update(L,v,k)`2.x = L`2.x \/ {G.order() -' k}
proof
  let G be _finite _Graph, L be LexBFS:Labeling of G, v be Vertex of G, x be
  set, k be Nat such that
A1: x in G.AdjacentSet({v}) and
A2: not x in dom L`1;
A3: x in (G.AdjacentSet({v}) \ dom L`1) by A1,A2,XBOOLE_0:def 5;
  then x in dom ((G.AdjacentSet({v}) \ dom L`1) --> {G.order()-'k});
  then
A4: x in dom (L`2) \/ dom ((G.AdjacentSet({v})\dom L`1)-->{G.order()-'k}) by
XBOOLE_0:def 3;
  set L2 = LexBFS:Update(L,v,k)`2;
  ((G.AdjacentSet({v}) \ dom L`1) --> {G.order()-'k}).x = {G.order()-'k}
  by A3,FUNCOP_1:7;
  hence thesis by A4,Def1;
end;
