let A, B be set ; :: thesis: ( ( for x being set holds
( x in A iff x is strict SubAlgebra of U0 ) ) & ( for x being set holds
( x in B iff x is strict SubAlgebra of U0 ) ) implies A = B )
assume A2: ( ( for x being set holds
( x in A iff x is strict SubAlgebra of U0 ) ) & ( for y being set holds
( y in B iff y is strict SubAlgebra of U0 ) ) ) ; :: thesis: A = B

now

let x be set ; :: thesis: ( x in A implies x in B )
assume x in A ; :: thesis: x in B
then x is strict SubAlgebra of U0 by A2;
hence x in B by A2; :: thesis: verum

end;

hence A c= B by TARSKI:def 3; :: according to XBOOLE_0:def 10 :: thesis: B c= A
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in B or y in A )
assume y in B ; :: thesis: y in A
then y is strict SubAlgebra of U0 by A2;
hence y in A by A2; :: thesis: verum