
theorem
  for V being RealLinearSpace, v1,v2,v3 being VECTOR of V, L being
  Linear_Combination of {v1,v2,v3} st v1 <> v2 & v2 <> v3 & v3 <> v1 & L is
convex holds L.v1 + L.v2 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 & Sum(L
  ) = L.v1 * v1 + L.v2 * v2 + L.v3 * v3
proof
  let V be RealLinearSpace, v1,v2,v3 be VECTOR of V, L be Linear_Combination
  of {v1,v2,v3};
  assume that
A1: v1 <> v2 and
A2: v2 <> v3 and
A3: v3 <> v1 and
A4: L is convex;
A5: Carrier(L) c= {v1,v2,v3} & Carrier(L) <> {} by A4,Th21,RLVECT_2:def 6;
  now
    per cases by A5,ZFMISC_1:118;
    suppose
A6:   Carrier(L) = {v1};
      then not v3 in Carrier(L) by A3,TARSKI:def 1;
      then not v3 in {v where v is Element of V : L.v <> 0} by RLVECT_2:def 4;
      then
A7:   L.v3 = 0;
      not v2 in Carrier(L) by A1,A6,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 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 by A4,A6,A7
,Lm11;
    end;
    suppose
A8:   Carrier(L) = {v2};
      then not v3 in Carrier(L) by A2,TARSKI:def 1;
      then not v3 in {v where v is Element of V : L.v <> 0} by RLVECT_2:def 4;
      then
A9:   L.v3 = 0;
      not v1 in Carrier(L) by A1,A8,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 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 by A4,A8
,A9,Lm11;
    end;
    suppose
A10:  Carrier(L) = {v3};
      then not v2 in Carrier(L) by A2,TARSKI:def 1;
      then not v2 in {v where v is Element of V : L.v <> 0} by RLVECT_2:def 4;
      then
A11:  L.v2 = 0;
      not v1 in Carrier(L) by A3,A10,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 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 by A4
,A10,A11,Lm11;
    end;
    suppose
A12:  Carrier(L) = {v1,v2};
      then not v3 in Carrier(L) by A2,A3,TARSKI:def 2;
      then not v3 in {v where v is Element of V : L.v <> 0} by RLVECT_2:def 4;
      then L.v3 = 0;
      hence
      L.v1 + L.v2 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 by A1,A4,A12
,Lm12;
    end;
    suppose
A13:  Carrier(L) = {v2,v3};
      then not v1 in Carrier(L) by A1,A3,TARSKI:def 2;
      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 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 by A2,A4,A13
,Lm12;
    end;
    suppose
A14:  Carrier(L) = {v1,v3};
      then not v2 in Carrier(L) by A1,A2,TARSKI:def 2;
      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 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 by A3,A4,A14
,Lm12;
    end;
    suppose
      Carrier(L) = {v1,v2,v3};
      hence
      L.v1 + L.v2 + L.v3 = 1 & L.v1 >= 0 & L.v2 >= 0 & L.v3 >= 0 by A1,A2,A3,A4
,Lm15;
    end;
  end;
  hence thesis by A1,A2,A3,Lm14;
end;
