
theorem Th1:
  for V being RealLinearSpace, v being VECTOR of V holds ex L being
Convex_Combination of V st Sum(L) = v & for A being non empty Subset of V st v
  in A holds L is Convex_Combination of A
proof
  let V be RealLinearSpace;
  let 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) by XBOOLE_0:def 10;
  then F = <*v*> by A2,FINSEQ_3:97;
  then
A4: F.1 = v by FINSEQ_1:def 8;
  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: 1 in REAL by XREAL_0:def 1;
A7: len F = 1 by A3,A2,FINSEQ_3:96;
  then 1 in dom f by A5,FINSEQ_3:25;
  then
A8: f.1 = L.(F.1) by A5;
  then f = <*1*> by A1,A5,A7,A4,FINSEQ_1:40;
  then rng f = {1} by FINSEQ_1:38;
  then rng f c= REAL by ZFMISC_1:31,A6;
  then reconsider f as FinSequence of REAL by FINSEQ_1:def 4;
A9: for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0
  proof
    let n be Nat;
    assume
A10: n in dom f;
    then n in Seg len f by FINSEQ_1:def 3;
    hence thesis by A1,A5,A7,A8,A4,A10,FINSEQ_1:2,TARSKI:def 1;
  end;
  f = <*1*> by A1,A5,A7,A8,A4,FINSEQ_1:40;
  then Sum(f) = jj by FINSOP_1:11;
  then reconsider L as Convex_Combination of V by A2,A5,A9,CONVEX1:def 3;
A11: for A being non empty Subset of V st v in A holds L is
  Convex_Combination of A by A3,RLVECT_2:def 6,ZFMISC_1:31;
  take L;
  Sum(L) = 1 * v by A1,A3,RLVECT_2:35;
  hence thesis by A11,RLVECT_1:def 8;
end;
