let V be ComplexLinearSpace; :: thesis: for v being VECTOR of V
for L being C_Linear_Combination of V st L is convex & Carrier L = {v} holds
( ex r being Real st
( r = L . v & r = 1 ) & Sum L = (L . v) * v )
let v be VECTOR of V; :: thesis: for L being C_Linear_Combination of V st L is convex & Carrier L = {v} holds
( ex r being Real st
( r = L . v & r = 1 ) & Sum L = (L . v) * v )
let L be C_Linear_Combination of V; :: thesis: ( L is convex & Carrier L = {v} implies ( ex r being Real st
( r = L . v & r = 1 ) & Sum L = (L . v) * v ) )
assume that
A1: L is convex and
A2: Carrier L = {v} ; :: thesis: ( ex r being Real st
( r = L . v & r = 1 ) & Sum L = (L . v) * v )
reconsider L = L as C_Linear_Combination of {v} by A2, Def4;
consider F being FinSequence of the carrier of V such that
A3: ( F is one-to-one & rng F = Carrier L ) and
A4: 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;
A5: F = <*v*> by A2, A3, FINSEQ_3:97;
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 A4;
reconsider r = f /. 1 as Element of REAL ;
card (Carrier L) = 1 by A2, CARD_1:30;
then len F = 1 by A3, FINSEQ_4:62;
then A9: dom f = Seg 1 by A6, FINSEQ_1:def 3;
then A10: 1 in dom f ;
then A11: f . 1 = f /. 1 by PARTFUN1:def 6;
then A12: f = <*r*> by A9, FINSEQ_1:def 8;
f . 1 = L . (F . 1) by A8, A10;
then r = L . v by A11, A5, FINSEQ_1:def 8;
hence ( ex r being Real st
( r = L . v & r = 1 ) & Sum L = (L . v) * v ) by A7, A12, Th14, FINSOP_1:11; :: thesis: verum