let V be RealLinearSpace; :: thesis: for v being VECTOR of V

for l being Linear_Combination of {v} holds Sum l = (l . v) * v

let v be VECTOR of V; :: thesis: for l being Linear_Combination of {v} holds Sum l = (l . v) * v

let l be Linear_Combination of {v}; :: thesis: Sum l = (l . v) * v

A1: Carrier l c= {v} by Def6;

for l being Linear_Combination of {v} holds Sum l = (l . v) * v

let v be VECTOR of V; :: thesis: for l being Linear_Combination of {v} holds Sum l = (l . v) * v

let l be Linear_Combination of {v}; :: thesis: Sum l = (l . v) * v

A1: Carrier l c= {v} by Def6;

now :: thesis: Sum l = (l . v) * vend;

hence
Sum l = (l . v) * v
; :: thesis: verumper cases
( Carrier l = {} or Carrier l = {v} )
by A1, ZFMISC_1:33;

end;

suppose
Carrier l = {}
; :: thesis: Sum l = (l . v) * v

then A2:
l = ZeroLC V
by Def5;

hence Sum l = 0. V by Lm2

.= 0 * v by RLVECT_1:10

.= (l . v) * v by A2, Th20 ;

:: thesis: verum

end;hence Sum l = 0. V by Lm2

.= 0 * v by RLVECT_1:10

.= (l . v) * v by A2, Th20 ;

:: thesis: verum

suppose
Carrier l = {v}
; :: thesis: Sum l = (l . v) * v

then consider F being FinSequence of V such that

A3: ( F is one-to-one & rng F = {v} ) and

A4: Sum l = Sum (l (#) F) by Def8;

F = <*v*> by A3, FINSEQ_3:97;

then l (#) F = <*((l . v) * v)*> by Th26;

hence Sum l = (l . v) * v by A4, RLVECT_1:44; :: thesis: verum

end;A3: ( F is one-to-one & rng F = {v} ) and

A4: Sum l = Sum (l (#) F) by Def8;

F = <*v*> by A3, FINSEQ_3:97;

then l (#) F = <*((l . v) * v)*> by Th26;

hence Sum l = (l . v) * v by A4, RLVECT_1:44; :: thesis: verum