
theorem
  for G being _finite EGraph, e, x being set st e in the_Edges_of G & not
e in G.labeledE() holds card G.labelEdge(e,x).labeledE() = card G.labeledE() +
  1
proof
  let G be _finite EGraph, e,val be set;
  set G2 = G.labelEdge(e,val);
  set ECurr = the_ELabel_of G, ENext = the_ELabel_of G2;
  assume e in the_Edges_of G & not e in G.labeledE();
  then
A1: card (dom ECurr \/ {e}) = card (dom ECurr) + 1 & ENext = ECurr +* (e
  .--> val) by Th32,CARD_2:41;
  dom (e.-->val) = {e};
  hence thesis by A1,FUNCT_4:def 1;
end;
