for b1 being set for b2 being set holds ( ( not b1 is empty implies ( b2 is Element of b1 iff b2in b1 ) ) & ( b1 is empty implies ( b2 is Element of b1 iff b2 is empty ) ) );
for b1 being set for b2, b3, b4 being Subset of b1 holds ( ( for b5 being Element of b1 holds ( b5in b2 iff ( b5in b3 or b5in b4 ) ) ) implies b2= b3\/ b4 )
for b1 being set for b2, b3, b4 being Subset of b1 holds ( ( for b5 being Element of b1 holds ( b5in b2 iff ( b5in b3 & b5in b4 ) ) ) implies b2= b3/\ b4 )
for b1 being set for b2, b3, b4 being Subset of b1 holds ( ( for b5 being Element of b1 holds ( b5in b2 iff ( b5in b3 & not b5in b4 ) ) ) implies b2= b3\ b4 )
for b1 being set for b2, b3, b4 being Subset of b1 holds ( ( for b5 being Element of b1 holds ( b5in b2 iff not ( b5in b3 iff b5in b4 ) ) ) implies b2= b3\+\ b4 )
for b1, b2 being Subset of F1() holds ( ( for b3 being Element of F1() holds ( b3in b1 iff P1[b3] ) ) & ( for b3 being Element of F1() holds ( b3in b2 iff P1[b3] ) ) implies b1= b2 )