let X be RealUnitarySpace; ( 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 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
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
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
; for x being Point of X
for S being finite OrthonormalFamily of X st not S is empty 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
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
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; for S being finite OrthonormalFamily of X st not S is empty 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
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
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; ( not S is empty 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
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
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
; 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
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
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; ( S c= dom H & ( for y being Point of X st y in S holds
H . y = (x .|. y) ^2 ) 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
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 H
and
A5:
for y being Point of X st y in S holds
H . y = (x .|. y) ^2
; 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
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; ( S c= dom F & ( for y being Point of X st y in S holds
F . y = (x .|. y) * y ) implies 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 F
and
A7:
for y being Point of X st y in S holds
F . y = (x .|. y) * y
; 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 y being Point of X st y in S holds
H . y = the scalar of X . [x,(F . y)]
A13:
setopfunc (S, the carrier of X,REAL,H,addreal) = addreal "**" (Func_Seq (H,p))
by A8, A9, Def5;
x .|. (setopfunc (S, the carrier of X, the carrier of X,F, the addF of X)) = the scalar of X . [x,( the addF of X "**" (Func_Seq (F,p)))]
by A10, BHSP_1:def 1;
hence
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)
by A3, A4, A6, A8, A9, A11, A13, Th10; verum