let V be RealLinearSpace; :: thesis: for W being Subspace of V
for L being Linear_Combination of V st Carrier L c= the carrier of W holds
ex K being Linear_Combination of W st
( Carrier K = Carrier L & Sum K = Sum L )
let W be Subspace of V; :: thesis: for L being Linear_Combination of V st Carrier L c= the carrier of W holds
ex K being Linear_Combination of W st
( Carrier K = Carrier L & Sum K = Sum L )
let L be Linear_Combination of V; :: thesis: ( Carrier L c= the carrier of W implies ex K being Linear_Combination of W st
( Carrier K = Carrier L & Sum K = Sum L ) )
assume A1: Carrier L c= the carrier of W ; :: thesis: ex K being Linear_Combination of W st
( Carrier K = Carrier L & Sum K = Sum L )
then reconsider C = Carrier L as finite Subset of W ;
the carrier of W c= the carrier of V by RLSUB_1:def 2;
then reconsider K = L | the carrier of W as Function of the carrier of W,REAL by FUNCT_2:38;
A2: dom K = the carrier of W by FUNCT_2:def 1;

A3: now

let w be VECTOR of W; :: thesis: ( not w in C implies K . w = 0 )
assume not w in C ; :: thesis: K . w = 0
then ( not w in C & w is VECTOR of V ) by RLSUB_1:18;
then L . w = 0 by RLVECT_2:28;
hence K . w = 0 by A2, FUNCT_1:70; :: thesis: verum

end;

K is Element of Funcs the carrier of W,REAL by FUNCT_2:11;
then reconsider K = K as Linear_Combination of W by A3, RLVECT_2:def 5;
take K ; :: thesis: ( Carrier K = Carrier L & Sum K = Sum L )
thus ( Carrier K = Carrier L & Sum K = Sum L ) by A1, Th11; :: thesis: verum