
theorem
  for G being EGraph, e1,e2,x being set st e1 <> e2 holds (the_ELabel_of
  G.labelEdge(e1,x)).e2 = (the_ELabel_of G).e2
proof
  let G be EGraph, e1,e2,x be set;
  set G2 = G.labelEdge(e1,x);
  assume
A1: e1 <> e2;
  now
    per cases;
    suppose
A2:   e1 in the_Edges_of G;
      not e2 in {e1} by A1,TARSKI:def 1;
      then
A3:   not e2 in dom (e1.-->x);
      the_ELabel_of G2 = the_ELabel_of G +* (e1.-->x) by A2,Th32;
      hence thesis by A3,FUNCT_4:11;
    end;
    suppose
      not e1 in the_Edges_of G;
      hence thesis by Def21;
    end;
  end;
  hence thesis;
end;
