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

( 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

X1 c= X2
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;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

proof

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