
theorem
  for G being VGraph, v1,v2,x being set st v1 <> v2 holds (the_VLabel_of
  G.labelVertex(v1,x)).v2 = (the_VLabel_of G).v2
proof
  let G be VGraph, v1,v2,x be set;
  set G2 = G.labelVertex(v1,x);
  assume
A1: v1 <> v2;
  now
    per cases;
    suppose
A2:   v1 in the_Vertices_of G;
      not v2 in {v1} by A1,TARSKI:def 1;
      then
A3:   not v2 in dom (v1.-->x);
      the_VLabel_of G2 = the_VLabel_of G +* (v1.-->x) by A2,Th38;
      hence thesis by A3,FUNCT_4:11;
    end;
    suppose
      not v1 in the_Vertices_of G;
      hence thesis by Def22;
    end;
  end;
  hence thesis;
end;
