let x be object ; :: thesis: for V being RealLinearSpace
for A being Subset of V st x in A holds
x in Lin A

let V be RealLinearSpace; :: thesis: for A being Subset of V st x in A holds
x in Lin A

let A be Subset of V; :: thesis: ( x in A implies x in Lin A )
deffunc H1( Element of V) -> Element of REAL = In (0,REAL);
assume A1: x in A ; :: thesis: x in Lin A
then reconsider v = x as VECTOR of V ;
consider f being Function of the carrier of V,REAL such that
A2: f . v = jj and
A3: for u being VECTOR of V st u <> v holds
f . u = H1(u) from reconsider f = f as Element of Funcs ( the carrier of V,REAL) by FUNCT_2:8;
ex T being finite Subset of V st
for u being VECTOR of V st not u in T holds
f . u = 0
proof
take T = {v}; :: thesis: for u being VECTOR of V st not u in T holds
f . u = 0

let u be VECTOR of V; :: thesis: ( not u in T implies f . u = 0 )
assume not u in T ; :: thesis: f . u = 0
then u <> v by TARSKI:def 1;
hence f . u = 0 by A3; :: thesis: verum
end;
then reconsider f = f as Linear_Combination of V by RLVECT_2:def 3;
A4: Carrier f c= {v}
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier f or x in {v} )
assume x in Carrier f ; :: thesis: x in {v}
then consider u being VECTOR of V such that
A5: x = u and
A6: f . u <> 0 ;
u = v by A3, A6;
hence x in {v} by ; :: thesis: verum
end;
then reconsider f = f as Linear_Combination of {v} by RLVECT_2:def 6;
A7: Sum f = 1 * v by
.= v by RLVECT_1:def 8 ;
{v} c= A by ;
then Carrier f c= A by A4;
then reconsider f = f as Linear_Combination of A by RLVECT_2:def 6;
Sum f = v by A7;
hence x in Lin A by Th14; :: thesis: verum