let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V holds Up (W1 /\ W2) = (Up W1) /\ (Up W2)
let W1, W2 be Subspace of V; :: thesis: Up (W1 /\ W2) = (Up W1) /\ (Up W2)
A1: Up (W1 /\ W2) = the carrier of (W1 /\ W2) by RUSUB_4:def 5
.= the carrier of W1 /\ the carrier of W2 by RLSUB_2:def 2 ;
for x being object st x in (Up W1) /\ (Up W2) holds
x in Up (W1 /\ W2)

proof

let x be object ; :: thesis: ( x in (Up W1) /\ (Up W2) implies x in Up (W1 /\ W2) )
assume A2: x in (Up W1) /\ (Up W2) ; :: thesis: x in Up (W1 /\ W2)
then x in Up W2 by XBOOLE_0:def 4;
then A3: x in the carrier of W2 by RUSUB_4:def 5;
x in Up W1 by A2, XBOOLE_0:def 4;
then x in the carrier of W1 by RUSUB_4:def 5;
hence x in Up (W1 /\ W2) by A1, A3, XBOOLE_0:def 4; :: thesis: verum

end;

then A4: (Up W1) /\ (Up W2) c= Up (W1 /\ W2) ;
for x being object st x in Up (W1 /\ W2) holds
x in (Up W1) /\ (Up W2)

proof

let x be object ; :: thesis: ( x in Up (W1 /\ W2) implies x in (Up W1) /\ (Up W2) )
assume A5: x in Up (W1 /\ W2) ; :: thesis: x in (Up W1) /\ (Up W2)
then x in the carrier of W2 by A1, XBOOLE_0:def 4;
then A6: x in Up W2 by RUSUB_4:def 5;
x in the carrier of W1 by A1, A5, XBOOLE_0:def 4;
then x in Up W1 by RUSUB_4:def 5;
hence x in (Up W1) /\ (Up W2) by A6, XBOOLE_0:def 4; :: thesis: verum

end;

then Up (W1 /\ W2) c= (Up W1) /\ (Up W2) ;
hence Up (W1 /\ W2) = (Up W1) /\ (Up W2) by A4, XBOOLE_0:def 10; :: thesis: verum