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 A2, RLVECT_2:def 6;

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 A2, A4, FINSEQ_4:62;

then A9: len f = 2 by A3, A6, CARD_2:57;

then A10: dom f = {1,2} by FINSEQ_1:2, FINSEQ_1:def 3;

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 A11, PARTFUN1:def 6;

then reconsider r1 = L . (F . 1) as Element of REAL ;

A13: 2 in dom f by A10, TARSKI:def 2;

then A14: f . 2 = L . (F . 2) by A8;

then f /. 2 = L . (F . 2) by A13, PARTFUN1:def 6;

then reconsider r2 = L . (F . 2) as Element of REAL ;

A15: f = <*r1,r2*> by A9, A12, A14, FINSEQ_1:44;

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 A2, RLVECT_2:def 6;

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 A2, A4, FINSEQ_4:62;

then A9: len f = 2 by A3, A6, CARD_2:57;

then A10: dom f = {1,2} by FINSEQ_1:2, FINSEQ_1:def 3;

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 A11, PARTFUN1:def 6;

then reconsider r1 = L . (F . 1) as Element of REAL ;

A13: 2 in dom f by A10, TARSKI:def 2;

then A14: f . 2 = L . (F . 2) by A8;

then f /. 2 = L . (F . 2) by A13, PARTFUN1:def 6;

then reconsider r2 = L . (F . 2) as Element of REAL ;

A15: f = <*r1,r2*> by A9, A12, A14, FINSEQ_1:44;

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

hence
( (L . v1) + (L . v2) = 1 & L . v1 >= 0 & L . v2 >= 0 & Sum L = ((L . v1) * v1) + ((L . v2) * v2) )
by A3, RLVECT_2:33; :: thesis: verumper cases
( F = <*v1,v2*> or F = <*v2,v1*> )
by A2, A3, A4, FINSEQ_3:99;

end;