
theorem Th61:
  for G being _finite _Graph, L be MCS:Labeling of G, v,x being set
  st x in dom L`1 holds L`2.x = (MCS:LabelAdjacent(L,v))`2.x
proof
  let G be _finite _Graph, L be MCS:Labeling of G, v,x be set such that
A1: x in dom L`1;
  set V2G = L`2;
  set VLG = L`1;
  set GL = MCS:LabelAdjacent(L,v);
  set V2 = GL`2;
  not x in G.AdjacentSet({v}) \ dom VLG by A1,XBOOLE_0:def 5;
  hence thesis by Def3;
end;
