let A, B be set ; :: thesis: ( ( for x being object holds

( x in A iff x is strict SubAlgebra of U0 ) ) & ( for x being object holds

( x in B iff x is strict SubAlgebra of U0 ) ) implies A = B )

assume that

A1: for x being object holds

( x in A iff x is strict SubAlgebra of U0 ) and

A2: for y being object holds

( y in B iff y is strict SubAlgebra of U0 ) ; :: thesis: A = B

let y be object ; :: 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 A1; :: thesis: verum

( x in A iff x is strict SubAlgebra of U0 ) ) & ( for x being object holds

( x in B iff x is strict SubAlgebra of U0 ) ) implies A = B )

assume that

A1: for x being object holds

( x in A iff x is strict SubAlgebra of U0 ) and

A2: for y being object holds

( y in B iff y is strict SubAlgebra of U0 ) ; :: thesis: A = B

now :: thesis: for x being object st x in A holds

x in B

hence
A c= B
; :: according to XBOOLE_0:def 10 :: thesis: B c= Ax in B

let x be object ; :: thesis: ( x in A implies x in B )

assume x in A ; :: thesis: x in B

then x is strict SubAlgebra of U0 by A1;

hence x in B by A2; :: thesis: verum

end;assume x in A ; :: thesis: x in B

then x is strict SubAlgebra of U0 by A1;

hence x in B by A2; :: thesis: verum

let y be object ; :: 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 A1; :: thesis: verum