defpred S₁[ set ] means ( $1 in X & ex f being Function of S,BOOLEAN st
( f = $1 & f . s = TRUE ) );
for X1, X2 being set st ( for x being set holds
( x in X1 iff S₁[x] ) ) & ( for x being set holds
( x in X2 iff S₁[x] ) ) holds
X1 = X2 from XBOOLE_0:sch 3();
hence for b₁, b₂ being Subset of X st ( for x being set holds
( x in b₁ iff ( x in X & ex f being Function of S,BOOLEAN st
( f = x & f . s = TRUE ) ) ) ) & ( for x being set holds
( x in b₂ iff ( x in X & ex f being Function of S,BOOLEAN st
( f = x & f . s = TRUE ) ) ) ) holds
b₁ = b₂ ; :: thesis: verum