
theorem
  for V being RealLinearSpace, v1,v2 being VECTOR of V, L being
Linear_Combination of {v1,v2} st v1 <> v2 & L is convex 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;
  let v1,v2 be VECTOR of V;
  let L be Linear_Combination of {v1,v2};
  assume that
A1: v1 <> v2 and
A2: L is convex;
A3: Carrier(L) c= {v1,v2} & Carrier(L) <> {} by A2,Th21,RLVECT_2:def 6;
  now
    per cases by A3,ZFMISC_1:36;
    suppose
A4:   Carrier(L) = {v1};
      then not v2 in Carrier(L) by A1,TARSKI:def 1;
      then not v2 in {v where v is Element of V : L.v <> 0} by RLVECT_2:def 4;
      then L.v2 = 0;
      hence L.v1 + L.v2 = 1 & L.v1 >= 0 & L.v2 >= 0 by A2,A4,Lm11;
    end;
    suppose
A5:   Carrier(L) = {v2};
      then not v1 in Carrier(L) by A1,TARSKI:def 1;
      then not v1 in {v where v is Element of V : L.v <> 0} by RLVECT_2:def 4;
      then L.v1 = 0;
      hence L.v1 + L.v2 = 1 & L.v1 >= 0 & L.v2 >= 0 by A2,A5,Lm11;
    end;
    suppose
      Carrier(L) = {v1,v2};
      hence L.v1 + L.v2 = 1 & L.v1 >= 0 & L.v2 >= 0 by A1,A2,Lm12;
    end;
  end;
  hence thesis by A1,RLVECT_2:33;
end;
