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).| (#) |.(seq_id x).| is summable ) ) ) & ( for x being object holds
( x in X2 iff ( x in the_set_of_ComplexSequences & |.(seq_id x).| (#) |.(seq_id x).| is summable ) ) ) implies X1 = X2 )
assume that
A2: for x being object holds
( x in X1 iff ( x in the_set_of_ComplexSequences & |.(seq_id x).| (#) |.(seq_id x).| is summable ) ) and
A3: for x being object holds
( x in X2 iff ( x in the_set_of_ComplexSequences & |.(seq_id x).| (#) |.(seq_id x).| is summable ) ) ; :: thesis: X1 = X2
A4: X2 c= X1

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X2 or x in X1 )
assume A5: x in X2 ; :: thesis: x in X1
then |.(seq_id x).| (#) |.(seq_id x).| is summable by A3;
hence x in X1 by A2, A5; :: thesis: verum

end;

X1 c= X2

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X1 or x in X2 )
assume A6: x in X1 ; :: thesis: x in X2
then |.(seq_id x).| (#) |.(seq_id x).| is summable by A2;
hence x in X2 by A3, A6; :: thesis: verum

end;

hence X1 = X2 by A4, XBOOLE_0:def 10; :: thesis: verum