let X1, X2 be Subset of Linear_Space_of_ComplexSequences; :: thesis: ( ( for x being object holds
( x in X1 iff ( x in the_set_of_ComplexSequences & seq_id x is bounded ) ) ) & ( for x being object holds
( x in X2 iff ( x in the_set_of_ComplexSequences & seq_id x is bounded ) ) ) implies X1 = X2 )

assume that
A2: for x being object holds
( x in X1 iff ( x in the_set_of_ComplexSequences & seq_id x is bounded ) ) and
A3: for x being object holds
( x in X2 iff ( x in the_set_of_ComplexSequences & seq_id x is bounded ) ) ; :: thesis: X1 = X2
thus X1 c= X2 :: according to XBOOLE_0:def 10 :: thesis: X2 c= X1
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X1 or x in X2 )
assume x in X1 ; :: thesis: x in X2
then ( x in the_set_of_ComplexSequences & seq_id x is bounded ) by A2;
hence x in X2 by A3; :: thesis: verum
end;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X2 or x in X1 )
assume x in X2 ; :: thesis: x in X1
then ( x in the_set_of_ComplexSequences & seq_id x is bounded ) by A3;
hence x in X1 by A2; :: thesis: verum