ex b1 being Equivalence_Relation of F1() st for b2, b3 being set holds ( [b2,b3]in b1 iff ( b2in F1() & b3in F1() & P1[b2,b3] ) )
provided
E11:
for b1 being set holds ( b1in F1() implies P1[b1,b1] )
and E12:
for b1, b2 being set holds ( P1[b1,b2] implies P1[b2,b1] )
and E13:
for b1, b2, b3 being set holds ( P1[b1,b2] & P1[b2,b3] implies P1[b1,b3] )
uniqueness
for b1, b2 being Subset-Family of c1 holds ( ( for b3 being Subset of c1 holds ( b3in b1 iff ex b4 being set st ( b4in c1 & b3=Class c2,b4 ) ) ) & ( for b3 being Subset of c1 holds ( b3in b2 iff ex b4 being set st ( b4in c1 & b3=Class c2,b4 ) ) ) implies b1= b2 )
for b1 being set for b2 being Subset-Family of b1 holds ( ( b1<>{} implies ( b2 is a_partition of b1 iff ( union b2= b1 & ( for b3 being Subset of b1 holds ( b3in b2 implies ( b3<>{} & ( for b4 being Subset of b1 holds not ( b4in b2 & not b3= b4 & not b3misses b4 ) ) ) ) ) ) ) ) & ( not b1<>{} implies ( b2 is a_partition of b1 iff b2={} ) ) );