let V be RealLinearSpace; :: thesis: for v1, v2 being VECTOR of V
for 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) )
let v1, v2 be VECTOR of V; :: thesis: for 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) )
let L be Linear_Combination of {v1,v2}; :: thesis: ( v1 <> v2 & L is convex implies ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) ) )
assume that
A1: v1 <> v2 and
A2: L is convex ; :: thesis: ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
A3: ( Carrier L c= {v1,v2} & Carrier L <> {} ) by A2, Th21, RLVECT_2:def 6;

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

per cases ( Carrier L = {v1} or Carrier L = {v2} or Carrier L = {v1,v2} ) by A3, ZFMISC_1:36;

suppose A4: Carrier L = {v1} ; :: thesis: ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 )

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; :: thesis: verum

end;

suppose A5: Carrier L = {v2} ; :: thesis: ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 )

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; :: thesis: verum

end;

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

hence ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 ) by A1, A2, Lm12; :: thesis: verum

end;

end;

end;

hence ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) ) by A1, RLVECT_2:33; :: thesis: verum