let n be Ordinal; :: thesis: for L being non trivial left_add-cancelable right_complementable add-associative right_zeroed well-unital distributive associative domRing-like left_zeroed doubleLoopStr

for p being Polynomial of n,L

for a being Element of L

for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let L be non trivial left_add-cancelable right_complementable add-associative right_zeroed well-unital distributive associative domRing-like left_zeroed doubleLoopStr ; :: thesis: for p being Polynomial of n,L

for a being Element of L

for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let p be Polynomial of n,L; :: thesis: for a being Element of L

for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let a be Element of L; :: thesis: for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let x be Function of n,L; :: thesis: eval ((a * p),x) = a * (eval (p,x))

consider y being FinSequence of the carrier of L such that

A1: len y = len (SgmX ((BagOrder n),(Support (a * p)))) and

A2: eval ((a * p),x) = Sum y and

A3: for i being Element of NAT st 1 <= i & i <= len y holds

y /. i = (((a * p) * (SgmX ((BagOrder n),(Support (a * p))))) /. i) * (eval (((SgmX ((BagOrder n),(Support (a * p)))) /. i),x)) by POLYNOM2:def 4;

A4: BagOrder n linearly_orders Support (a * p) by POLYNOM2:18;

consider z being FinSequence of the carrier of L such that

A5: len z = len (SgmX ((BagOrder n),(Support p))) and

A6: eval (p,x) = Sum z and

A7: for i being Element of NAT st 1 <= i & i <= len z holds

z /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x)) by POLYNOM2:def 4;

for p being Polynomial of n,L

for a being Element of L

for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let L be non trivial left_add-cancelable right_complementable add-associative right_zeroed well-unital distributive associative domRing-like left_zeroed doubleLoopStr ; :: thesis: for p being Polynomial of n,L

for a being Element of L

for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let p be Polynomial of n,L; :: thesis: for a being Element of L

for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let a be Element of L; :: thesis: for x being Function of n,L holds eval ((a * p),x) = a * (eval (p,x))

let x be Function of n,L; :: thesis: eval ((a * p),x) = a * (eval (p,x))

consider y being FinSequence of the carrier of L such that

A1: len y = len (SgmX ((BagOrder n),(Support (a * p)))) and

A2: eval ((a * p),x) = Sum y and

A3: for i being Element of NAT st 1 <= i & i <= len y holds

y /. i = (((a * p) * (SgmX ((BagOrder n),(Support (a * p))))) /. i) * (eval (((SgmX ((BagOrder n),(Support (a * p)))) /. i),x)) by POLYNOM2:def 4;

A4: BagOrder n linearly_orders Support (a * p) by POLYNOM2:18;

consider z being FinSequence of the carrier of L such that

A5: len z = len (SgmX ((BagOrder n),(Support p))) and

A6: eval (p,x) = Sum z and

A7: for i being Element of NAT st 1 <= i & i <= len z holds

z /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x)) by POLYNOM2:def 4;

per cases
( a = 0. L or a <> 0. L )
;

end;

suppose A8:
a = 0. L
; :: thesis: eval ((a * p),x) = a * (eval (p,x))

then SgmX ((BagOrder n),(Support (a * p))) = {} by RELAT_1:41;

then y = <*> the carrier of L by A1;

then Sum y = 0. L by RLVECT_1:43

.= a * (Sum z) by A8 ;

hence eval ((a * p),x) = a * (eval (p,x)) by A2, A6; :: thesis: verum

end;

A9: now :: thesis: for b being bag of n holds (a * p) . b = 0. L

let b be bag of n; :: thesis: (a * p) . b = 0. L

thus (a * p) . b = a * (p . b) by Def9

.= 0. L by A8 ; :: thesis: verum

end;thus (a * p) . b = a * (p . b) by Def9

.= 0. L by A8 ; :: thesis: verum

now :: thesis: not Support (a * p) <> {}

then
rng (SgmX ((BagOrder n),(Support (a * p)))) = {}
by A4, PRE_POLY:def 2;assume
Support (a * p) <> {}
; :: thesis: contradiction

then reconsider sp = Support (a * p) as non empty Subset of (Bags n) ;

set c = the Element of sp;

(a * p) . the Element of sp <> 0. L by POLYNOM1:def 4;

hence contradiction by A9; :: thesis: verum

end;then reconsider sp = Support (a * p) as non empty Subset of (Bags n) ;

set c = the Element of sp;

(a * p) . the Element of sp <> 0. L by POLYNOM1:def 4;

hence contradiction by A9; :: thesis: verum

then SgmX ((BagOrder n),(Support (a * p))) = {} by RELAT_1:41;

then y = <*> the carrier of L by A1;

then Sum y = 0. L by RLVECT_1:43

.= a * (Sum z) by A8 ;

hence eval ((a * p),x) = a * (eval (p,x)) by A2, A6; :: thesis: verum

suppose A10:
a <> 0. L
; :: thesis: eval ((a * p),x) = a * (eval (p,x))

A11:
for u being object st u in Support (a * p) holds

u in Support p

u in Support (a * p)

then A16: dom z = Seg (len y) by FINSEQ_1:def 3

.= dom y by FINSEQ_1:def 3 ;

A17: Support (a * p) = Support p by A13, A11, TARSKI:2;

hence eval ((a * p),x) = a * (eval (p,x)) by A2, A6, BINOM:4; :: thesis: verum

end;u in Support p

proof

A13:
for u being object st u in Support p holds
let u be object ; :: thesis: ( u in Support (a * p) implies u in Support p )

assume A12: u in Support (a * p) ; :: thesis: u in Support p

then reconsider u9 = u as Element of Bags n ;

(a * p) . u <> 0. L by A12, POLYNOM1:def 4;

then a * (p . u9) <> 0. L by Def9;

then p . u9 <> 0. L ;

hence u in Support p by POLYNOM1:def 4; :: thesis: verum

end;assume A12: u in Support (a * p) ; :: thesis: u in Support p

then reconsider u9 = u as Element of Bags n ;

(a * p) . u <> 0. L by A12, POLYNOM1:def 4;

then a * (p . u9) <> 0. L by Def9;

then p . u9 <> 0. L ;

hence u in Support p by POLYNOM1:def 4; :: thesis: verum

u in Support (a * p)

proof

then A15:
len z = len y
by A1, A5, A11, TARSKI:2;
let u be object ; :: thesis: ( u in Support p implies u in Support (a * p) )

assume A14: u in Support p ; :: thesis: u in Support (a * p)

then reconsider u9 = u as Element of Bags n ;

p . u <> 0. L by A14, POLYNOM1:def 4;

then a * (p . u9) <> 0. L by A10, VECTSP_2:def 1;

then (a * p) . u9 <> 0. L by Def9;

hence u in Support (a * p) by POLYNOM1:def 4; :: thesis: verum

end;assume A14: u in Support p ; :: thesis: u in Support (a * p)

then reconsider u9 = u as Element of Bags n ;

p . u <> 0. L by A14, POLYNOM1:def 4;

then a * (p . u9) <> 0. L by A10, VECTSP_2:def 1;

then (a * p) . u9 <> 0. L by Def9;

hence u in Support (a * p) by POLYNOM1:def 4; :: thesis: verum

then A16: dom z = Seg (len y) by FINSEQ_1:def 3

.= dom y by FINSEQ_1:def 3 ;

A17: Support (a * p) = Support p by A13, A11, TARSKI:2;

now :: thesis: for i being object st i in dom z holds

y /. i = a * (z /. i)

then
y = a * z
by A16, POLYNOM1:def 1;y /. i = a * (z /. i)

A18:
dom (a * p) = Bags n
by FUNCT_2:def 1;

then reconsider r = (a * p) * (SgmX ((BagOrder n),(Support (a * p)))) as FinSequence by FINSEQ_1:16;

for u being object st u in rng r holds

u in the carrier of L

A21: dom p = Bags n by FUNCT_2:def 1;

then reconsider q = p * (SgmX ((BagOrder n),(Support (a * p)))) as FinSequence by FINSEQ_1:16;

for u being object st u in rng q holds

u in the carrier of L

reconsider r = r as FinSequence of the carrier of L by A20, FINSEQ_1:def 4;

reconsider q = q as FinSequence of the carrier of L by A23, FINSEQ_1:def 4;

let i be object ; :: thesis: ( i in dom z implies y /. i = a * (z /. i) )

assume A24: i in dom z ; :: thesis: y /. i = a * (z /. i)

then i in Seg (len z) by FINSEQ_1:def 3;

then i in { k where k is Nat : ( 1 <= k & k <= len z ) } by FINSEQ_1:def 1;

then consider k being Nat such that

A25: i = k and

A26: ( 1 <= k & k <= len z ) ;

reconsider k = k as Element of NAT by ORDINAL1:def 12;

A27: dom z = Seg (len (SgmX ((BagOrder n),(Support (a * p))))) by A1, A16, FINSEQ_1:def 3

.= dom (SgmX ((BagOrder n),(Support (a * p)))) by FINSEQ_1:def 3 ;

then (SgmX ((BagOrder n),(Support (a * p)))) . k = (SgmX ((BagOrder n),(Support (a * p)))) /. k by A24, A25, PARTFUN1:def 6;

then k in dom q by A24, A25, A27, A21, FUNCT_1:11;

then A28: (p * (SgmX ((BagOrder n),(Support (a * p))))) /. k = q . k by PARTFUN1:def 6

.= p . ((SgmX ((BagOrder n),(Support (a * p)))) . k) by A24, A25, A27, FUNCT_1:13

.= p . ((SgmX ((BagOrder n),(Support (a * p)))) /. k) by A24, A25, A27, PARTFUN1:def 6 ;

reconsider c = (SgmX ((BagOrder n),(Support (a * p)))) /. k as Element of Bags n ;

reconsider c = c as bag of n ;

A29: a * (z /. k) = a * (((p * (SgmX ((BagOrder n),(Support p)))) /. k) * (eval (((SgmX ((BagOrder n),(Support p))) /. k),x))) by A7, A26

.= (a * ((p * (SgmX ((BagOrder n),(Support (a * p))))) /. k)) * (eval (((SgmX ((BagOrder n),(Support (a * p)))) /. k),x)) by A17, GROUP_1:def 3 ;

A30: (a * p) . ((SgmX ((BagOrder n),(Support (a * p)))) /. k) = a * ((p * (SgmX ((BagOrder n),(Support (a * p))))) /. k) by A28, Def9;

(SgmX ((BagOrder n),(Support (a * p)))) . k = (SgmX ((BagOrder n),(Support (a * p)))) /. k by A24, A25, A27, PARTFUN1:def 6;

then k in dom r by A24, A25, A27, A18, FUNCT_1:11;

then ((a * p) * (SgmX ((BagOrder n),(Support (a * p))))) /. k = r . k by PARTFUN1:def 6

.= (a * p) . ((SgmX ((BagOrder n),(Support (a * p)))) . k) by A24, A25, A27, FUNCT_1:13

.= a * ((p * (SgmX ((BagOrder n),(Support (a * p))))) /. k) by A24, A25, A27, A30, PARTFUN1:def 6 ;

hence y /. i = a * (z /. i) by A3, A15, A25, A26, A29; :: thesis: verum

end;now :: thesis: for u being object st u in rng (SgmX ((BagOrder n),(Support (a * p)))) holds

u in dom (a * p)

then
rng (SgmX ((BagOrder n),(Support (a * p)))) c= dom (a * p)
by TARSKI:def 3;u in dom (a * p)

let u be object ; :: thesis: ( u in rng (SgmX ((BagOrder n),(Support (a * p)))) implies u in dom (a * p) )

assume u in rng (SgmX ((BagOrder n),(Support (a * p)))) ; :: thesis: u in dom (a * p)

then u in Support (a * p) by A4, PRE_POLY:def 2;

hence u in dom (a * p) by A18; :: thesis: verum

end;assume u in rng (SgmX ((BagOrder n),(Support (a * p)))) ; :: thesis: u in dom (a * p)

then u in Support (a * p) by A4, PRE_POLY:def 2;

hence u in dom (a * p) by A18; :: thesis: verum

then reconsider r = (a * p) * (SgmX ((BagOrder n),(Support (a * p)))) as FinSequence by FINSEQ_1:16;

for u being object st u in rng r holds

u in the carrier of L

proof

then A20:
rng r c= the carrier of L
by TARSKI:def 3;
let u be object ; :: thesis: ( u in rng r implies u in the carrier of L )

assume u in rng r ; :: thesis: u in the carrier of L

then A19: u in rng (a * p) by FUNCT_1:14;

rng (a * p) c= the carrier of L by RELAT_1:def 19;

hence u in the carrier of L by A19; :: thesis: verum

end;assume u in rng r ; :: thesis: u in the carrier of L

then A19: u in rng (a * p) by FUNCT_1:14;

rng (a * p) c= the carrier of L by RELAT_1:def 19;

hence u in the carrier of L by A19; :: thesis: verum

A21: dom p = Bags n by FUNCT_2:def 1;

now :: thesis: for u being object st u in rng (SgmX ((BagOrder n),(Support (a * p)))) holds

u in dom p

then
rng (SgmX ((BagOrder n),(Support (a * p)))) c= dom p
by TARSKI:def 3;u in dom p

let u be object ; :: thesis: ( u in rng (SgmX ((BagOrder n),(Support (a * p)))) implies u in dom p )

assume u in rng (SgmX ((BagOrder n),(Support (a * p)))) ; :: thesis: u in dom p

then u in Support (a * p) by A4, PRE_POLY:def 2;

hence u in dom p by A21; :: thesis: verum

end;assume u in rng (SgmX ((BagOrder n),(Support (a * p)))) ; :: thesis: u in dom p

then u in Support (a * p) by A4, PRE_POLY:def 2;

hence u in dom p by A21; :: thesis: verum

then reconsider q = p * (SgmX ((BagOrder n),(Support (a * p)))) as FinSequence by FINSEQ_1:16;

for u being object st u in rng q holds

u in the carrier of L

proof

then A23:
rng q c= the carrier of L
by TARSKI:def 3;
let u be object ; :: thesis: ( u in rng q implies u in the carrier of L )

assume u in rng q ; :: thesis: u in the carrier of L

then A22: u in rng p by FUNCT_1:14;

rng p c= the carrier of L by RELAT_1:def 19;

hence u in the carrier of L by A22; :: thesis: verum

end;assume u in rng q ; :: thesis: u in the carrier of L

then A22: u in rng p by FUNCT_1:14;

rng p c= the carrier of L by RELAT_1:def 19;

hence u in the carrier of L by A22; :: thesis: verum

reconsider r = r as FinSequence of the carrier of L by A20, FINSEQ_1:def 4;

reconsider q = q as FinSequence of the carrier of L by A23, FINSEQ_1:def 4;

let i be object ; :: thesis: ( i in dom z implies y /. i = a * (z /. i) )

assume A24: i in dom z ; :: thesis: y /. i = a * (z /. i)

then i in Seg (len z) by FINSEQ_1:def 3;

then i in { k where k is Nat : ( 1 <= k & k <= len z ) } by FINSEQ_1:def 1;

then consider k being Nat such that

A25: i = k and

A26: ( 1 <= k & k <= len z ) ;

reconsider k = k as Element of NAT by ORDINAL1:def 12;

A27: dom z = Seg (len (SgmX ((BagOrder n),(Support (a * p))))) by A1, A16, FINSEQ_1:def 3

.= dom (SgmX ((BagOrder n),(Support (a * p)))) by FINSEQ_1:def 3 ;

then (SgmX ((BagOrder n),(Support (a * p)))) . k = (SgmX ((BagOrder n),(Support (a * p)))) /. k by A24, A25, PARTFUN1:def 6;

then k in dom q by A24, A25, A27, A21, FUNCT_1:11;

then A28: (p * (SgmX ((BagOrder n),(Support (a * p))))) /. k = q . k by PARTFUN1:def 6

.= p . ((SgmX ((BagOrder n),(Support (a * p)))) . k) by A24, A25, A27, FUNCT_1:13

.= p . ((SgmX ((BagOrder n),(Support (a * p)))) /. k) by A24, A25, A27, PARTFUN1:def 6 ;

reconsider c = (SgmX ((BagOrder n),(Support (a * p)))) /. k as Element of Bags n ;

reconsider c = c as bag of n ;

A29: a * (z /. k) = a * (((p * (SgmX ((BagOrder n),(Support p)))) /. k) * (eval (((SgmX ((BagOrder n),(Support p))) /. k),x))) by A7, A26

.= (a * ((p * (SgmX ((BagOrder n),(Support (a * p))))) /. k)) * (eval (((SgmX ((BagOrder n),(Support (a * p)))) /. k),x)) by A17, GROUP_1:def 3 ;

A30: (a * p) . ((SgmX ((BagOrder n),(Support (a * p)))) /. k) = a * ((p * (SgmX ((BagOrder n),(Support (a * p))))) /. k) by A28, Def9;

(SgmX ((BagOrder n),(Support (a * p)))) . k = (SgmX ((BagOrder n),(Support (a * p)))) /. k by A24, A25, A27, PARTFUN1:def 6;

then k in dom r by A24, A25, A27, A18, FUNCT_1:11;

then ((a * p) * (SgmX ((BagOrder n),(Support (a * p))))) /. k = r . k by PARTFUN1:def 6

.= (a * p) . ((SgmX ((BagOrder n),(Support (a * p)))) . k) by A24, A25, A27, FUNCT_1:13

.= a * ((p * (SgmX ((BagOrder n),(Support (a * p))))) /. k) by A24, A25, A27, A30, PARTFUN1:def 6 ;

hence y /. i = a * (z /. i) by A3, A15, A25, A26, A29; :: thesis: verum

hence eval ((a * p),x) = a * (eval (p,x)) by A2, A6, BINOM:4; :: thesis: verum