
theorem
  for G being VGraph, v1,v2,x being set st not v2 in G.labeledV() & v2
  in G.labelVertex(v1,x).labeledV() holds v1 = v2 & v1 in the_Vertices_of G
proof
  let G be VGraph, e1,e2,val be set;
  set Gn = G.labelVertex(e1,val);
  assume that
A1: not e2 in G.labeledV() and
A2: e2 in Gn.labeledV();
  e1 in the_Vertices_of G by A1,A2,Def22;
  then the_VLabel_of Gn = (the_VLabel_of G) +* (e1 .--> val) by Th38;
  then e2 in dom (the_VLabel_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,Def22;
end;
