theorem
  v|still_not-bound_in p = w|still_not-bound_in p implies v.(x|a)|
  still_not-bound_in p = w.(x|a)|still_not-bound_in p
proof
  assume
A1: v|still_not-bound_in p = w|still_not-bound_in p;
  dom (v|still_not-bound_in p) = dom (v.(x|a)|still_not-bound_in p) by Th63;
  then
A2: dom (v.(x|a)|still_not-bound_in p) = dom (w.(x|a)|still_not-bound_in p)
  by A1,Th63;
  for b being object
st b in dom (v.(x|a)|still_not-bound_in p) holds (v.(x|a)|
  still_not-bound_in p).b = (w.(x|a)|still_not-bound_in p).b
  proof
    let b being object such that
A3: b in dom (v.(x|a)|still_not-bound_in p);
A4: (v.(x|a)|still_not-bound_in p).b = v.(x|a).b & (w.(x|a)|
    still_not-bound_in p ).b = w.(x|a).b by A2,A3,FUNCT_1:47;
    b in dom (v|still_not-bound_in p) by A3,Th63;
    then
A5: (v|still_not-bound_in p).b = v.b by FUNCT_1:47;
    b in dom (w|still_not-bound_in p) by A1,A3,Th63;
    then
A6: v.b = w.b by A1,A5,FUNCT_1:47;
A7: now
      assume
A8:   b <> x;
      then v.(x|a).b = v.b by Th48;
      hence thesis by A4,A6,A8,Th48;
    end;
    now
      assume
A9:   b = x;
      then v.(x|a).b = a by Th49;
      hence thesis by A4,A9,Th49;
    end;
    hence thesis by A7;
  end;
  hence thesis by A2,FUNCT_1:2;
end;
