
theorem Th62:
  for G being _finite _Graph, L being MCS:Labeling of G, v,x being
  set st x in dom L`2 & x in G.AdjacentSet{v} & not x in dom L`1 holds (
  MCS:LabelAdjacent(L,v))`2.x = (L`2).x + 1
proof
  let G be _finite _Graph, L being MCS:Labeling of G, v,x be set such that
A1: x in dom L`2 and
A2: x in G.AdjacentSet({v}) and
A3: not x in dom L`1;
  set V2G = L`2;
  set VLG = L`1;
  set GL = MCS:LabelAdjacent(L,v);
  set V2 = GL`2;
  x in G.AdjacentSet({v}) \ dom VLG by A2,A3,XBOOLE_0:def 5;
  hence thesis by A1,Def3;
end;
