let X be RealUnitarySpace; :: thesis: ( the addF of X is commutative & the addF of X is associative & the addF of X is having_a_unity implies for x being Point of X
for S being finite OrthonormalFamily of X st not S is empty holds
for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y being Point of X st y in S holds
F . y = (x .|. y) * y ) holds
for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) holds
(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal) )
assume that
A1: ( the addF of X is commutative & the addF of X is associative ) and
A2: the addF of X is having_a_unity ; :: thesis: for x being Point of X
for S being finite OrthonormalFamily of X st not S is empty holds
for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y being Point of X st y in S holds
F . y = (x .|. y) * y ) holds
for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) holds
(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
let x be Point of X; :: thesis: for S being finite OrthonormalFamily of X st not S is empty holds
for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y being Point of X st y in S holds
F . y = (x .|. y) * y ) holds
for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) holds
(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
let S be finite OrthonormalFamily of X; :: thesis: ( not S is empty implies for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y being Point of X st y in S holds
F . y = (x .|. y) * y ) holds
for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) holds
(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal) )
assume A3: not S is empty ; :: thesis: for F being Function of the carrier of X, the carrier of X st S c= dom F & ( for y being Point of X st y in S holds
F . y = (x .|. y) * y ) holds
for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) holds
(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
let F be Function of the carrier of X, the carrier of X; :: thesis: ( S c= dom F & ( for y being Point of X st y in S holds
F . y = (x .|. y) * y ) implies for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) holds
(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal) )
assume that
A4: S c= dom F and
A5: for y being Point of X st y in S holds
F . y = (x .|. y) * y ; :: thesis: for H being Function of the carrier of X,REAL st S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) holds
(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
let H be Function of the carrier of X,REAL; :: thesis: ( S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) implies (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal) )
assume that
A6: S c= dom H and
A7: for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ; :: thesis: (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal)
consider p being FinSequence of the carrier of X such that
A8: p is one-to-one and
A9: rng p = S and
A10: setopfunc (S, the carrier of X, the carrier of X,F, the addF of X) = the addF of X "**" (Func_Seq (F,p)) by A1, A2, Def5;
A11: for y1, y2 being Point of X st y1 in S & y2 in S & y1 <> y2 holds
the scalar of X . [(F . y1),(F . y2)] = 0

proof

let y1, y2 be Point of X; :: thesis: ( y1 in S & y2 in S & y1 <> y2 implies the scalar of X . [(F . y1),(F . y2)] = 0 )
assume that
A12: y1 in S and
A13: y2 in S and
A14: y1 <> y2 ; :: thesis: the scalar of X . [(F . y1),(F . y2)] = 0
set z1 = x .|. y1;
set z2 = x .|. y2;
S is OrthogonalFamily of X by Def9;
then A15: y1 .|. y2 = 0 by A12, A13, A14, Def8;
the scalar of X . [(F . y1),(F . y2)] = the scalar of X . [((x .|. y1) * y1),(F . y2)] by A5, A12
.= the scalar of X . [((x .|. y1) * y1),((x .|. y2) * y2)] by A5, A13
.= ((x .|. y1) * y1) .|. ((x .|. y2) * y2) by BHSP_1:def 1
.= (x .|. y2) * (y2 .|. ((x .|. y1) * y1)) by BHSP_1:3
.= (x .|. y2) * ((x .|. y1) * (y2 .|. y1)) by BHSP_1:3
.= 0 by A15 ;
hence the scalar of X . [(F . y1),(F . y2)] = 0 ; :: thesis: verum

end;

A16: for y being Point of X st y in S holds
H . y = the scalar of X . [(F . y),(F . y)]

proof

let y be Point of X; :: thesis: ( y in S implies H . y = the scalar of X . [(F . y),(F . y)] )
assume A17: y in S ; :: thesis: H . y = the scalar of X . [(F . y),(F . y)]
then A18: F . y = (x .|. y) * y by A5;
H . y = (x .|. y) ^2 by A7, A17
.= ((x .|. y) * (x .|. y)) * 1
.= ((x .|. y) * (x .|. y)) * (y .|. y) by A17, Def9
.= (x .|. y) * ((x .|. y) * (y .|. y))
.= (x .|. y) * (((x .|. y) * y) .|. y) by BHSP_1:3
.= ((x .|. y) * y) .|. ((x .|. y) * y) by BHSP_1:3
.= the scalar of X . [(F . y),(F . y)] by A18, BHSP_1:def 1 ;
hence H . y = the scalar of X . [(F . y),(F . y)] ; :: thesis: verum

end;

(setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = the scalar of X . [( the addF of X "**" (Func_Seq (F,p))),( the addF of X "**" (Func_Seq (F,p)))] by A10, BHSP_1:def 1
.= addreal "**" (Func_Seq (H,p)) by A3, A4, A6, A8, A9, A11, A16, Th9
.= setopfunc (S, the carrier of X,REAL,H,addreal) by A8, A9, Def5 ;
hence (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = setopfunc (S, the carrier of X,REAL,H,addreal) ; :: thesis: verum