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
.= { (v + u) where u, v is VECTOR of V : ( v in W1 & u in W2 ) } by RLSUB_2:def 1 ;
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 x in (Up W1) + (Up W2) ; :: thesis: x in Up (W1 + W2)
then x in { (u + v) where u, v is Element of V : ( u in Up W1 & v in Up W2 ) } by RUSUB_4:def 9;
then consider u, v being Element of V such that
A2: x = u + v and
A3: u in Up W1 and
A4: v in Up W2 ;
v in the carrier of W2 by A4, RUSUB_4:def 5;
then A5: v in W2 by STRUCT_0:def 5;
u in the carrier of W1 by A3, RUSUB_4:def 5;
then u in W1 by STRUCT_0:def 5;
hence x in Up (W1 + W2) by A1, A2, A5; :: thesis: verum

end;

then A6: (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 x in Up (W1 + W2) ; :: thesis: x in (Up W1) + (Up W2)
then consider u, v being VECTOR of V such that
A7: x = v + u and
A8: v in W1 and
A9: u in W2 by A1;
u in the carrier of W2 by A9, STRUCT_0:def 5;
then A10: u in Up W2 by RUSUB_4:def 5;
v in the carrier of W1 by A8, STRUCT_0:def 5;
then v in Up W1 by RUSUB_4:def 5;
then x in { (u9 + v9) where u9, v9 is Element of V : ( u9 in Up W1 & v9 in Up W2 ) } by A7, A10;
hence x in (Up W1) + (Up W2) by RUSUB_4:def 9; :: thesis: verum

end;

then Up (W1 + W2) c= (Up W1) + (Up W2) ;
hence Up (W1 + W2) = (Up W1) + (Up W2) by A6, XBOOLE_0:def 10; :: thesis: verum