
theorem Th45:
  for G being EGraph, e,x being set st e in the_Edges_of G holds G
  .labelEdge(e,x).labeledE() = G.labeledE() \/ {e}
proof
  let G be EGraph, e,val be set;
  set G2 = G.labelEdge(e,val), EG = the_ELabel_of G, EG2 = the_ELabel_of G2;
  assume e in the_Edges_of G;
  then EG2 = EG +* (e.--> val) by Th32;
  then dom EG2 = dom EG \/ dom (e.-->val) by FUNCT_4:def 1;
  hence thesis;
end;
