
theorem
  for G being VGraph, v,x being set st v in the_Vertices_of G holds (
  the_VLabel_of G.labelVertex(v,x)).v = x
proof
  let G be VGraph, e,x be set;
  e in {e} by TARSKI:def 1;
  then
A1: e in dom (e.-->x);
  assume e in the_Vertices_of G;
  then the_VLabel_of G.labelVertex(e,x) = the_VLabel_of G +* (e.-->x) by Th38;
  then (the_VLabel_of G.labelVertex(e,x)).e = (e.-->x).e by A1,FUNCT_4:13
    .= x by FUNCOP_1:72;
  hence thesis;
end;
