let A1, A2 be a_partition of Y; :: thesis: ( ( for d being set holds

( d in A1 iff d is_min_depend PA,PB ) ) & ( for d being set holds

( d in A2 iff d is_min_depend PA,PB ) ) implies A1 = A2 )

assume that

A26: for x being set holds

( x in A1 iff x is_min_depend PA,PB ) and

A27: for x being set holds

( x in A2 iff x is_min_depend PA,PB ) ; :: thesis: A1 = A2

for y being object holds

( y in A1 iff y in A2 ) by A26, A27;

hence A1 = A2 by TARSKI:2; :: thesis: verum

( d in A1 iff d is_min_depend PA,PB ) ) & ( for d being set holds

( d in A2 iff d is_min_depend PA,PB ) ) implies A1 = A2 )

assume that

A26: for x being set holds

( x in A1 iff x is_min_depend PA,PB ) and

A27: for x being set holds

( x in A2 iff x is_min_depend PA,PB ) ; :: thesis: A1 = A2

for y being object holds

( y in A1 iff y in A2 ) by A26, A27;

hence A1 = A2 by TARSKI:2; :: thesis: verum