
theorem Th17:
  for V being RealLinearSpace, L being Linear_Combination of V, v
  being VECTOR of V st L is circled & L.v <= 0 holds not v in Carrier(L)
proof
  let V be RealLinearSpace, L be Linear_Combination of V, v be VECTOR of V;
  assume that
A1: L is circled and
A2: L.v <= 0;
  per cases by A2;
  suppose
    L.v = 0;
    hence thesis by RLVECT_2:19;
  end;
  suppose
A3: L.v < 0;
    now
      consider F being FinSequence of the carrier of V such that
      F is one-to-one and
A4:   rng F = Carrier L and
A5:   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;
      assume v in Carrier(L);
      then consider n be object such that
A6:   n in dom F and
A7:   F.n = v by A4,FUNCT_1:def 3;
      reconsider n as Element of NAT by A6;
      consider f being FinSequence of REAL such that
A8:   len f = len F and
      Sum f = 1 and
A9:   for n being Nat st n in dom f holds f.n = L.(F.n) & f.n >= 0 by A5;
      n in Seg len F by A6,FINSEQ_1:def 3;
      then
A10:  n in dom f by A8,FINSEQ_1:def 3;
      then L.v = f.n by A9,A7;
      hence contradiction by A3,A9,A10;
    end;
    hence thesis;
  end;
end;
