let X1, X2 be Element of F_{1}(); :: thesis: ( ( for x being object holds

( x in X1 iff P_{1}[x] ) ) & ( for x being object holds

( x in X2 iff P_{1}[x] ) ) implies X1 = X2 )

assume that

A1: for x being object holds

( x in X1 iff P_{1}[x] )
and

A2: for x being object holds

( x in X2 iff P_{1}[x] )
; :: thesis: X1 = X2

for x being object holds

( x in X1 iff x in X2 ) by A1, A2;

hence X1 = X2 by TARSKI:2; :: thesis: verum

( x in X1 iff P

( x in X2 iff P

assume that

A1: for x being object holds

( x in X1 iff P

A2: for x being object holds

( x in X2 iff P

for x being object holds

( x in X1 iff x in X2 ) by A1, A2;

hence X1 = X2 by TARSKI:2; :: thesis: verum