
theorem
  for V being RealLinearSpace, v being VECTOR of V holds ex L being
circled_Combination of V st Sum L = v & for A being non empty Subset of V st v
  in A holds L is circled_Combination of A
proof
  let V be RealLinearSpace, v be VECTOR of V;
  consider L being Linear_Combination of {v} such that
A1: L.v = jj by RLVECT_4:37;
  consider F being FinSequence of the carrier of V such that
A2: F is one-to-one & rng F = Carrier(L) and
  Sum(L) = Sum(L (#) F) by RLVECT_2:def 8;
  v in Carrier L by A1,RLVECT_2:19;
  then Carrier L c= {v} & {v} c= Carrier L by RLVECT_2:def 6,ZFMISC_1:31;
  then
A3: {v} = Carrier L;
  then F = <*v*> by A2,FINSEQ_3:97;
  then
A4: F.1 = v;
  deffunc F(set) = L.(F.$1);
  consider f being FinSequence such that
A5: len f = len F & for n being Nat st n in dom f holds f.n = F(n) from
  FINSEQ_1:sch 2;
A6: len F = 1 by A3,A2,FINSEQ_3:96;
  then 1 in dom f by A5,FINSEQ_3:25;
  then
A7: f.1 = L.(F.1) by A5;
  then f = <*jj*> by A1,A5,A6,A4,FINSEQ_1:40;
  then reconsider f as FinSequence of REAL;
A8: for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
  proof
    let n be Nat;
    assume
A9: n in dom f;
    then n in Seg len f by FINSEQ_1:def 3;
    hence thesis by A1,A5,A6,A7,A4,A9,FINSEQ_1:2,TARSKI:def 1;
  end;
  f = <*jj*> by A1,A5,A6,A7,A4,FINSEQ_1:40;
  then Sum(f) = 1 by FINSOP_1:11;
  then reconsider L as circled_Combination of V by A2,A5,A8,Def4;
A10: for A being non empty Subset of V st v in A holds L is
  circled_Combination of A
  by ZFMISC_1:31,A3,RLVECT_2:def 6;
  take L;
  Sum(L) = 1 * v by A1,A3,RLVECT_2:35;
  hence thesis by A10,RLVECT_1:def 8;
end;
