
theorem Th22:
  for V being RealLinearSpace, v being VECTOR of V, L being
Linear_Combination of V st L is circled & Carrier(L) = {v} holds L.v = 1 & Sum(
  L) = L.v * v
proof
  let V be RealLinearSpace, v be VECTOR of V, L be Linear_Combination of V;
  assume that
A1: L is circled and
A2: Carrier(L) = {v};
  reconsider L as Linear_Combination of {v} by A2,RLVECT_2:def 6;
  consider F being FinSequence of the carrier of V such that
A3: F is one-to-one & rng F = Carrier L and
A4: ex f being FinSequence of REAL st len f = len F & Sum(f) = 1 & for n
  being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0 by A1;
A5: F = <*v*> by A2,A3,FINSEQ_3:97;
  consider f be FinSequence of REAL such that
A6: len f = len F and
A7: Sum(f) = 1 and
A8: for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0 by A4;
  reconsider r = f/.1 as Element of REAL;
  card Carrier(L) = 1 by A2,CARD_1:30;
  then len F = 1 by A3,FINSEQ_4:62;
  then
A9: dom f = Seg 1 by A6,FINSEQ_1:def 3;
  then
A10: 1 in dom f by FINSEQ_1:2,TARSKI:def 1;
  then
A11: f.1 = f/.1 by PARTFUN1:def 6;
  then f = <* r *> by A9,FINSEQ_1:def 8;
  then
A12: Sum f = r by FINSOP_1:11;
  f.1 = L.(F.1) by A8,A10;
  hence thesis by A7,A11,A12,A5,RLVECT_2:32;
end;
