
theorem Th26:
  for G being _finite _Graph, L being LexBFS:Labeling of G, v being
Vertex of G, x being set, k being Nat st x in dom L`1 holds LexBFS:Update(L,v,k
  )`2.x = L`2.x
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 dom L`1;
  set F = (G.AdjacentSet({v}) \ dom L`1) --> {G.order()-'k};
A2: not x in dom F by A1,XBOOLE_0:def 5;
  then
A3: F.x = {} by FUNCT_1:def 2;
  set L2 = LexBFS:Update(L,v,k)`2;
  per cases;
  suppose
    x in dom L`2;
    then x in dom L`2 \/ dom F by XBOOLE_0:def 3;
    hence L2.x = L`2.x \/ F.x by Def1
      .= L`2.x by A3;
  end;
  suppose
A4: not x in dom L`2;
    then not x in dom L`2 \/ dom F by A2,XBOOLE_0:def 3;
    then not x in dom L2 by Def1;
    hence L2.x = {} by FUNCT_1:def 2
      .= L`2.x by A4,FUNCT_1:def 2;
  end;
end;
