let X1, X2 be Subset of Linear_Space_of_RealSequences; :: thesis: ( ( for x being object holds
( x in X1 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) ) & ( for x being object holds
( x in X2 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) ) implies X1 = X2 )
assume that
A2: for x being object holds
( x in X1 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) and
A3: for x being object holds
( x in X2 iff ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) ) ; :: thesis: X1 = X2
for x being object st x in X2 holds
x in X1

proof

let x be object ; :: thesis: ( x in X2 implies x in X1 )
assume x in X2 ; :: thesis: x in X1
then ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) by A3;
hence x in X1 by A2; :: thesis: verum

end;

then A4: X2 c= X1 by TARSKI:def 3;
for x being object st x in X1 holds
x in X2

proof

let x be object ; :: thesis: ( x in X1 implies x in X2 )
assume x in X1 ; :: thesis: x in X2
then ( x in the_set_of_RealSequences & seq_id x is absolutely_summable ) by A2;
hence x in X2 by A3; :: thesis: verum

end;

then X1 c= X2 by TARSKI:def 3;
hence X1 = X2 by A4, XBOOLE_0:def 10; :: thesis: verum