let X1, X2 be Subset of Linear_Space_of_ComplexSequences; :: thesis: ( not X1 is empty & ( for x being set holds
( x in X1 iff ( x in the_set_of_ComplexSequences & |.(seq_id x).| (#) |.(seq_id x).| is summable ) ) ) & not X2 is empty & ( for x being set holds
( x in X2 iff ( x in the_set_of_ComplexSequences & |.(seq_id x).| (#) |.(seq_id x).| is summable ) ) ) implies X1 = X2 )
assume that
not X1 is empty and
A3: for x being set holds
( x in X1 iff ( x in the_set_of_ComplexSequences & |.(seq_id x).| (#) |.(seq_id x).| is summable ) ) and
not X2 is empty and
A4: for x being set holds
( x in X2 iff ( x in the_set_of_ComplexSequences & |.(seq_id x).| (#) |.(seq_id x).| is summable ) ) ; :: thesis: X1 = X2
A5: X2 c= X1

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X2 or x in X1 )
assume A6: x in X2 ; :: thesis: x in X1
then |.(seq_id x).| (#) |.(seq_id x).| is summable by A4;
hence x in X1 by A3, A6; :: thesis: verum

end;

X1 c= X2

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in X1 or x in X2 )
assume A7: x in X1 ; :: thesis: x in X2
then |.(seq_id x).| (#) |.(seq_id x).| is summable by A3;
hence x in X2 by A4, A7; :: thesis: verum

end;

hence X1 = X2 by A5, XBOOLE_0:def 10; :: thesis: verum