
theorem
  for G being EGraph, e,x being set st e in the_Edges_of G holds (
  the_ELabel_of G.labelEdge(e,x)).e = x
proof
  let G be EGraph, e,x be set;
  e in {e} by TARSKI:def 1;
  then
A1: e in dom (e.-->x);
  assume e in the_Edges_of G;
  then the_ELabel_of G.labelEdge(e,x) = the_ELabel_of G +* (e.-->x) by Th32;
  then (the_ELabel_of G.labelEdge(e,x)).e = (e.-->x).e by A1,FUNCT_4:13
    .= x by FUNCOP_1:72;
  hence thesis;
end;
