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) (#) (seq_id x) is summable ) ) ) & ( for x being object holds

( x in X2 iff ( x in the_set_of_RealSequences & (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_RealSequences & (seq_id x) (#) (seq_id x) is summable ) ) and

A3: for x being object holds

( x in X2 iff ( x in the_set_of_RealSequences & (seq_id x) (#) (seq_id x) is summable ) ) ; :: thesis: X1 = X2

thus X1 c= X2 :: according to XBOOLE_0:def 10 :: thesis: X2 c= 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

( x in X1 iff ( x in the_set_of_RealSequences & (seq_id x) (#) (seq_id x) is summable ) ) ) & ( for x being object holds

( x in X2 iff ( x in the_set_of_RealSequences & (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_RealSequences & (seq_id x) (#) (seq_id x) is summable ) ) and

A3: for x being object holds

( x in X2 iff ( x in the_set_of_RealSequences & (seq_id x) (#) (seq_id x) is summable ) ) ; :: thesis: X1 = X2

thus X1 c= X2 :: according to XBOOLE_0:def 10 :: thesis: X2 c= X1

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X2 or x in X1 )
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X1 or x in X2 )

assume A4: x in X1 ; :: thesis: x in X2

then (seq_id x) (#) (seq_id x) is summable by A2;

hence x in X2 by A3, A4; :: thesis: verum

end;assume A4: x in X1 ; :: thesis: x in X2

then (seq_id x) (#) (seq_id x) is summable by A2;

hence x in X2 by A3, A4; :: thesis: verum

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