
theorem Th52:
  for G being VGraph, v,x being set st v in the_Vertices_of G
  holds G.labelVertex(v,x).labeledV() = G.labeledV() \/ {v}
proof
  let G be VGraph, e,val be set;
  set G2 = G.labelVertex(e,val),EG=the_VLabel_of G,EG2=the_VLabel_of G2;
  assume e in the_Vertices_of G;
  then EG2 = EG +* (e.--> val) by Th38;
  then dom EG2 = dom EG \/ dom (e.-->val) by FUNCT_4:def 1;
  hence thesis;
end;
