D). On the other hand, it follows from Ip - { P 1 1 2 that Ip & Ip - ~ (since/5 ~ - . . ALL, P2 - f (P 1)} ALL & ['i ~ a). 3. 4. Rules for making complex interpretations To build an interpretation of a complex formula means to define the variable's values providing consistency of a given formula. Consider the next formula vx(3y(S(x,y)& M(y)) ~ (3z(I(~)&(Ex, z))). By sequential transformations we obtain Vx(S(x, f (x))& M ( f (x)) --+ I('~)& E(x,5))~ Vx(S(x, f (x))& M ( f (x)) v l(5)& E(x,5)) => 3 x ( S ( x , f (x))& M ( f (x))&(I(5)v E(x,5))) S(~x,f(Sl))& M(f(Sl))& (Z(e) v E(B ,~)).

Let us consider one complete example illustrating the proof on the basis of the atomic interpretations. An example. Let us prove that the formula = 3 z ( P ( a , z )) may be derived from q, - v x 3 y ( P ( x , y)). First of all, we create an interpretation for the formula ~p 9 (i)~o = 'V'x3 y ( P ( x , y ) ) zz:>Vx P(x,f (Pl)) ~ 3x P ( x , f (P1)) => ~P(D,f(P1)) where P 1 is a free variable. Then we create an interpretation for the formula 9 9 Problem classification. Introduction to the solving methods 49 3z P(a,z) ~ P(a,D ) (ii) where "a" is a symbolic constant.