set u = the Element of V;
consider L being Linear_Combination of { the Element of V} such that
A1: L . the Element of V = jj by RLVECT_4:37;
take L ; :: thesis: L is circled
L is circled

proof

take <* the Element of V*> ; :: according to CIRCLED1:def 4 :: thesis: ( <* the Element of V*> is one-to-one & rng <* the Element of V*> = Carrier L & ex f being FinSequence of REAL st
( len f = len <* the Element of V*> & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (<* the Element of V*> . n) & f . n >= 0 ) ) ) )
A2: ex f being FinSequence of REAL st
( len f = len <* the Element of V*> & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (<* the Element of V*> . n) & f . n >= 0 ) ) )

proof

reconsider f = <*jj*> as FinSequence of REAL ;
take f ; :: thesis: ( len f = len <* the Element of V*> & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (<* the Element of V*> . n) & f . n >= 0 ) ) )
A3: for n being Nat st n in dom f holds
( f . n = L . (<* the Element of V*> . n) & f . n >= 0 )

proof

let n be Nat; :: thesis: ( n in dom f implies ( f . n = L . (<* the Element of V*> . n) & f . n >= 0 ) )
assume n in dom f ; :: thesis: ( f . n = L . (<* the Element of V*> . n) & f . n >= 0 )
then n in {1} by FINSEQ_1:2, FINSEQ_1:38;
then A4: n = 1 by TARSKI:def 1;
then f . n = L . the Element of V by A1
.= L . (<* the Element of V*> . n) by A4 ;
hence ( f . n = L . (<* the Element of V*> . n) & f . n >= 0 ) by A4; :: thesis: verum

end;

len <*jj*> = 1 by FINSEQ_1:39
.= len <* the Element of V*> by FINSEQ_1:39 ;
hence ( len f = len <* the Element of V*> & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (<* the Element of V*> . n) & f . n >= 0 ) ) ) by A3, FINSOP_1:11; :: thesis: verum

end;

the Element of V in { w where w is Element of V : L . w <> 0 } by A1;
then the Element of V in Carrier L by RLVECT_2:def 4;
then ( Carrier L c= { the Element of V} & { the Element of V} c= Carrier L ) by RLVECT_2:def 6, ZFMISC_1:31;
then Carrier L = { the Element of V} ;
hence ( <* the Element of V*> is one-to-one & rng <* the Element of V*> = Carrier L & ex f being FinSequence of REAL st
( len f = len <* the Element of V*> & Sum f = 1 & ( for n being Nat st n in dom f holds
( f . n = L . (<* the Element of V*> . n) & f . n >= 0 ) ) ) ) by A2, FINSEQ_1:38, FINSEQ_3:93; :: thesis: verum

end;

hence L is circled ; :: thesis: verum