
theorem
  for G being _finite VGraph, v, x being set st v in the_Vertices_of G &
  not v in G.labeledV() holds card G.labelVertex(v,x).labeledV() = card G
  .labeledV() + 1
proof
  let G be _finite VGraph, e,val be set;
  set G2 = G.labelVertex(e,val);
  set ECurr = the_VLabel_of G, ENext = the_VLabel_of G2;
  assume e in the_Vertices_of G & not e in G.labeledV();
  then
A1: card (dom ECurr \/ {e}) = card (dom ECurr) + 1 & ENext = ECurr +* (e
  .--> val) by Th38,CARD_2:41;
  dom (e.-->val) = {e};
  hence thesis by A1,FUNCT_4:def 1;
end;
