
theorem
  for G being EGraph, e1,e2,x being set st not e2 in G.labeledE() & e2
  in G.labelEdge(e1,x).labeledE() holds e1 = e2 & e1 in the_Edges_of G
proof
  let G be EGraph, e1,e2,val be set;
  set Gn = G.labelEdge(e1,val);
  assume that
A1: not e2 in G.labeledE() and
A2: e2 in Gn.labeledE();
  e1 in the_Edges_of G by A1,A2,Def21;
  then the_ELabel_of Gn = (the_ELabel_of G) +* (e1 .--> val) by Th32;
  then e2 in dom (the_ELabel_of G) or e2 in dom (e1 .--> val) by A2,FUNCT_4:12;
  then e2 in {e1} by A1;
  hence e1 = e2 by TARSKI:def 1;
  thus thesis by A1,A2,Def21;
end;
