
theorem Th23:
  for V being RealLinearSpace, v1,v2 being VECTOR of V, L being
Linear_Combination of V st L is circled & Carrier(L) = {v1,v2} & v1 <> v2 holds
  L.v1 + L.v2 = 1 & L.v1 >= 0 & L.v2 >= 0 & Sum(L) = L.v1 * v1 + L.v2 * v2
proof
  let V be RealLinearSpace, v1,v2 be VECTOR of V, L be Linear_Combination of V;
  assume that
A1: L is circled and
A2: Carrier(L) = {v1,v2} and
A3: v1 <> v2;
  reconsider L as Linear_Combination of {v1,v2} by A2,RLVECT_2:def 6;
  consider F being FinSequence of the carrier of V such that
A4: F is one-to-one & 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;
  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 A5;
  len F = card {v1,v2} by A2,A4,FINSEQ_4:62;
  then
A9: len f = 2 by A3,A6,CARD_2:57;
  then
A10: dom f = {1,2} by FINSEQ_1:2,def 3;
  then
A11: 1 in dom f by TARSKI:def 2;
  then
A12: f.1 = L.(F.1) by A8;
  then f/.1 = L.(F.1) by A11,PARTFUN1:def 6;
  then reconsider r1 = L.(F.1) as Element of REAL;
A13: 2 in dom f by A10,TARSKI:def 2;
  then
A14: f.2 = L.(F.2) by A8;
  then f/.2 = L.(F.2) by A13,PARTFUN1:def 6;
  then reconsider r2 = L.(F.2) as Element of REAL;
A15: f = <*r1,r2*> by A9,A12,A14,FINSEQ_1:44;
  now
    per cases by A2,A3,A4,FINSEQ_3:99;
    suppose
      F = <*v1,v2*>;
      then F.1 = v1 & F.2 = v2;
      hence L.v1 + L.v2 = 1 & L.v1 >= 0 & L.v2 >= 0 by A7,A8,A11,A13,A12,A14
,A15,RVSUM_1:77;
    end;
    suppose
      F = <*v2,v1*>;
      then F.1 = v2 & F.2 = v1;
      hence L.v1 + L.v2 = 1 & L.v1 >= 0 & L.v2 >= 0 by A7,A8,A11,A13,A12,A14
,A15,RVSUM_1:77;
    end;
  end;
  hence thesis by A3,RLVECT_2:33;
end;
