theorem Th66:
  v|(still_not-bound_in p \ {x}) = w|(still_not-bound_in p \ {x})
  implies v.(x|a)|still_not-bound_in p = w.(x|a)|still_not-bound_in p
proof
A1: dom (w.(x|a)|still_not-bound_in p) = 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 Th63;
  assume
A3: v|(still_not-bound_in p \ {x}) = w|(still_not-bound_in p \ {x});
  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
A4: b in dom (v.(x|a)|still_not-bound_in p);
A5: (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,A4,FUNCT_1:47;
A6: now
      assume
A7:   b <> x;
      then
A8:   not b in {x} by TARSKI:def 1;
      b in still_not-bound_in p by A4,Th63;
      then
A9:   b in still_not-bound_in p \ {x} by A8,XBOOLE_0:def 5;
      then b in dom (w|(still_not-bound_in p \ {x})) by Th63;
      then
A10:  w|(still_not-bound_in p \ {x}).b = w.b by FUNCT_1:47;
A11:  v.(x|a).b = v.b & w.(x|a).b = w.b by A7,Th48;
      b in dom (v|(still_not-bound_in p \ {x})) by A9,Th63;
      hence thesis by A3,A5,A10,A11,FUNCT_1:47;
    end;
    now
A12:  w.(x|a)|(still_not-bound_in p).b = w.(x|a).b by A2,A4,FUNCT_1:47;
      assume
A13:  b = x;
      v.(x|a)|(still_not-bound_in p).b = v.(x|a).b by A4,FUNCT_1:47;
      then v.(x|a)|(still_not-bound_in p).b = a by A13,Th49;
      hence thesis by A13,A12,Th49;
    end;
    hence thesis by A6;
  end;
  hence thesis by A1,Th63,FUNCT_1:2;
end;
