let V be RealLinearSpace; :: thesis: for v1, v2, v3 being VECTOR of V
for 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) )
let v1, v2, v3 be VECTOR of V; :: thesis: for 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) )
let L be Linear_Combination of {v1,v2,v3}; :: thesis: ( v1 <> v2 & v2 <> v3 & v3 <> v1 & L is convex implies ( ((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) ) )
assume that
A1: v1 <> v2 and
A2: v2 <> v3 and
A3: v3 <> v1 and
A4: L is convex ; :: thesis: ( ((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) )
A5: ( Carrier L c= {v1,v2,v3} & Carrier L <> {} ) by A4, Th21, RLVECT_2:def 6;

now :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

per cases ( Carrier L = {v1} or Carrier L = {v2} or Carrier L = {v3} or Carrier L = {v1,v2} or Carrier L = {v2,v3} or Carrier L = {v1,v3} or Carrier L = {v1,v2,v3} ) by A5, ZFMISC_1:118;

suppose A6: Carrier L = {v1} ; :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

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; :: thesis: verum

end;

suppose A8: Carrier L = {v2} ; :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

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; :: thesis: verum

end;

suppose A10: Carrier L = {v3} ; :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

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; :: thesis: verum

end;

suppose A12: Carrier L = {v1,v2} ; :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

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; :: thesis: verum

end;

suppose A13: Carrier L = {v2,v3} ; :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

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; :: thesis: verum

end;

suppose A14: Carrier L = {v1,v3} ; :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

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; :: thesis: verum

end;

suppose Carrier L = {v1,v2,v3} ; :: thesis: ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 )

hence ( ((L . v1) + (L . v2)) + (L . v3) = 1 & L . v1 >= 0 & L . v2 >= 0 & L . v3 >= 0 ) by A1, A2, A3, A4, Lm15; :: thesis: verum

end;

end;

end;

hence ( ((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) ) by A1, A2, A3, Lm14; :: thesis: verum