let V be RealLinearSpace; :: thesis: for v1, v2 being VECTOR of V
for L being Linear_Combination of V st L is convex & 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) )

let v1, v2 be VECTOR of V; :: thesis: for L being Linear_Combination of V st L is convex & 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) )

let L be Linear_Combination of V; :: thesis: ( L is convex & Carrier L = {v1,v2} & v1 <> v2 implies ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) ) )
assume that
A1: L is convex and
A2: Carrier L = {v1,v2} and
A3: v1 <> v2 ; :: thesis: ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
reconsider L = L as Linear_Combination of {v1,v2} by ;
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 being 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 ;
then A9: len f = 2 by ;
then A10: dom f = {1,2} by ;
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 ;
then reconsider r1 = L . (F . 1) as Element of REAL ;
A13: 2 in dom f by ;
then A14: f . 2 = L . (F . 2) by A8;
then f /. 2 = L . (F . 2) by ;
then reconsider r2 = L . (F . 2) as Element of REAL ;
A15: f = <*r1,r2*> by ;
now :: thesis: ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 )
per cases ( F = <*v1,v2*> or F = <*v2,v1*> ) by ;
suppose F = <*v1,v2*> ; :: thesis: ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 )
then ( F . 1 = v1 & F . 2 = v2 ) by FINSEQ_1:44;
hence ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 ) by ; :: thesis: verum
end;
suppose F = <*v2,v1*> ; :: thesis: ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 )
then ( F . 1 = v2 & F . 2 = v1 ) by FINSEQ_1:44;
hence ( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 ) by ; :: 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 ; :: thesis: verum