( ), . , . () , , ().
, ( ), , , , .
, :
1) - P Q;
2) - P Q;
3) - P Q;
4) - P Q.
:
1) x (P (x) → Q (x)); ( - x, x P ( P (x) - ), Q ( Q (x) - ); - x, P, Q);
2) x (P (x) Q (x));
3) x (P (x) → ┐ Q (x));
4) x (P (x) ┐ Q (x)).
, , - , .
.
1.
, :
f (x)=0. 䒺. , f (x)=0 䒺.
.
, : P (x) x , Q (x) x
f (x)=0, R (x, y) x y, S (x) x - 䒺.
:
x (P (x) y (Q (y) → R (x, y))),
z (P (z) → u (S (u) → ┐ R (z, u))),
w (Q (w) → ┐ S (w)).
,
, .
. ,
x (P (x) y (Q (y) R (x, y)))
:
x (P (x) y (┐ Q (y) ˅ R (x, y)))
( ┐ ˅ );
x y (P (x) ( ┐ Q (y) ˅ R (x, y)))
( y);
y (P (a) ( ┐ Q (y) ˅ R (a, y))) (1)
(a - ).
:
z (P (z) u (S (u) ┐ R (z, u)));
z ( ┐ P (z) ˅ u ( ┐ S (u) ˅┐ R (z, u)))
|
|
( ┐ ˅ );
z u ( ┐ P (z) ˅┐ S (u) ˅┐ R (z, u)) (2)
( u).
,
┐ w (Q (w) ┐ S (w))
. , :
┐ w (┐ Q (w) ˅ ┐ S (w));
w ┐(┐Q (w) ˅ ┐S (w));
w (Q (w) S (w));
Q (b) S (b). (3)
b -
(1), (2), (3), :
1. P (a);
2. ┐ Q (y) ˅ R (a, y);
3. ┐ P (z) ˅ ┐ S (u) ┐ R (z, u);
4. Q (b);
5. S (b),
.
, :
6. R (a, b) (4 2; ={ y:= b });
7. ┐ P (a) ˅ ┐ S (b) (6 3; ={ z:= a; u:= b });
8. ┐P (a) (7 5);
9. □ (8 1)..
, , , . , .