let X1, X2 be non empty Subset of Linear_Space_of_RealSequences; :: thesis: ( ( for x being set holds
( x in X1 iff ( x in the_set_of_RealSequences & (seq_id x) rto_power p is summable ) ) ) & ( for x being set holds
( x in X2 iff ( x in the_set_of_RealSequences & (seq_id x) rto_power p is summable ) ) ) implies X1 = X2 )
assume that
A4: for x being set holds
( x in X1 iff ( x in the_set_of_RealSequences & (seq_id x) rto_power p is summable ) ) and
A5: for x being set holds
( x in X2 iff ( x in the_set_of_RealSequences & (seq_id x) rto_power p is summable ) ) ; :: thesis: X1 = X2
A6: X2 c= X1

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X2 or x in X1 )
assume A7: x in X2 ; :: thesis: x in X1
then A8: (seq_id x) rto_power p is summable by A5;
x in the_set_of_RealSequences by A7;
hence x in X1 by A4, A8; :: 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 A9: x in X1 ; :: thesis: x in X2
then A10: (seq_id x) rto_power p is summable by A4;
x in the_set_of_RealSequences by A9;
hence x in X2 by A5, A10; :: thesis: verum

end;

hence X1 = X2 by A6, XBOOLE_0:def 10; :: thesis: verum