let A1, A2 be set ; :: thesis: ( ( for x being set holds
( x in A1 iff x is a_partition of Y ) ) & ( for x being set holds
( x in A2 iff x is a_partition of Y ) ) implies A1 = A2 )
assume that
A3: for x being set holds
( x in A1 iff x is a_partition of Y ) and
A4: for x being set holds
( x in A2 iff x is a_partition of Y ) ; :: thesis: A1 = A2

now

let y be set ; :: thesis: ( y in A1 iff y in A2 )
( y in A1 iff y is a_partition of Y ) by A3;
hence ( y in A1 iff y in A2 ) by A4; :: thesis: verum

end;

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