let n be Ordinal; :: thesis: for L being non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for c being ConstPoly of n,L
for x being Function of n,L holds eval c,x = coefficient c
let L be non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; :: thesis: for c being ConstPoly of n,L
for x being Function of n,L holds eval c,x = coefficient c
let c be ConstPoly of n,L; :: thesis: for x being Function of n,L holds eval c,x = coefficient c
let x be Function of n,L; :: thesis: eval c,x = coefficient c
consider y being FinSequence of the carrier of L such that
A1: len y = len (SgmX (BagOrder n),(Support c)) and
A2: eval c,x = Sum y and
A3: for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((c * (SgmX (BagOrder n),(Support c))) /. i) * (eval (((SgmX (BagOrder n),(Support c)) /. i) @ ),x) by POLYNOM2:def 4;

now

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

case A4: coefficient c = 0. L ; :: thesis: eval c,x = coefficient c

Support c = {}

proof

consider u being Element of Support c;
assume A5: Support c <> {} ; :: thesis: contradiction
then u in Support c ;
then reconsider u = u as Element of Bags n ;
c . u <> 0. L by A5, POLYNOM1:def 9;
then u <> EmptyBag n by A4, Lm2;
then c . u = 0. L by Def8;
hence contradiction by A5, POLYNOM1:def 9; :: thesis: verum

end;

then reconsider Sc = Support c as empty Subset of (Bags n) ;
SgmX (BagOrder n),Sc = {} by POLYNOM2:9, POLYNOM2:20;
then y = <*> the carrier of L by A1;
hence eval c,x = coefficient c by A2, A4, RLVECT_1:60; :: thesis: verum

end;

case A6: coefficient c <> 0. L ; :: thesis: eval c,x = coefficient c

reconsider sc = Support c as finite Subset of (Bags n) ;
set sg = SgmX (BagOrder n),sc;
A7: BagOrder n linearly_orders sc by POLYNOM2:20;
A8: for u being set st u in Support c holds
u in {(EmptyBag n)}

proof

let u be set ; :: thesis: ( u in Support c implies u in {(EmptyBag n)} )
assume A9: u in Support c ; :: thesis: u in {(EmptyBag n)}
assume A10: not u in {(EmptyBag n)} ; :: thesis: contradiction
reconsider u = u as Element of Bags n by A9;
u <> EmptyBag n by A10, TARSKI:def 1;
then c . u = 0. L by Def8;
hence contradiction by A9, POLYNOM1:def 9; :: thesis: verum

end;

for u being set st u in {(EmptyBag n)} holds
u in Support c

proof

let u be set ; :: thesis: ( u in {(EmptyBag n)} implies u in Support c )
assume A11: u in {(EmptyBag n)} ; :: thesis: u in Support c
then A12: u = EmptyBag n by TARSKI:def 1;
reconsider u = u as Element of Bags n by A11;
c . u <> 0. L by A6, A12, Lm2;
hence u in Support c by POLYNOM1:def 9; :: thesis: verum

end;

then Support c = {(EmptyBag n)} by A8, TARSKI:2;
then A13: rng (SgmX (BagOrder n),sc) = {(EmptyBag n)} by A7, PRE_POLY:def 2;
then A14: EmptyBag n in rng (SgmX (BagOrder n),sc) by TARSKI:def 1;
then A15: 1 in dom (SgmX (BagOrder n),sc) by FINSEQ_3:33;
then A16: (SgmX (BagOrder n),sc) . 1 in rng (SgmX (BagOrder n),sc) by FUNCT_1:12;
A17: for u being set st u in dom (SgmX (BagOrder n),sc) holds
u in {1}

proof

let u be set ; :: thesis: ( u in dom (SgmX (BagOrder n),sc) implies u in {1} )
assume A18: u in dom (SgmX (BagOrder n),sc) ; :: thesis: u in {1}
assume A19: not u in {1} ; :: thesis: contradiction
reconsider u = u as Element of NAT by A18;
(SgmX (BagOrder n),sc) /. u = (SgmX (BagOrder n),sc) . u by A18, PARTFUN1:def 8;
then A20: (SgmX (BagOrder n),sc) /. u in rng (SgmX (BagOrder n),sc) by A18, FUNCT_1:12;
A21: u <> 1 by A19, TARSKI:def 1;
A22: 1 < u

proof

consider k being Nat such that
A23: dom (SgmX (BagOrder n),sc) = Seg k by FINSEQ_1:def 2;
Seg k = { m where m is Element of NAT : ( 1 <= m & m <= k ) } by FINSEQ_1:def 1;
then ex m9 being Element of NAT st
( m9 = u & 1 <= m9 & m9 <= k ) by A18, A23;
hence 1 < u by A21, XXREAL_0:1; :: thesis: verum

end;

(SgmX (BagOrder n),sc) /. 1 = (SgmX (BagOrder n),sc) . 1 by A14, A18, FINSEQ_3:33, PARTFUN1:def 8;
then (SgmX (BagOrder n),sc) /. 1 in rng (SgmX (BagOrder n),sc) by A15, FUNCT_1:12;
then (SgmX (BagOrder n),sc) /. 1 = EmptyBag n by A13, TARSKI:def 1
.= (SgmX (BagOrder n),sc) /. u by A13, A20, TARSKI:def 1 ;
hence contradiction by A7, A15, A18, A22, PRE_POLY:def 2; :: thesis: verum

end;

for u being set st u in {1} holds
u in dom (SgmX (BagOrder n),sc) by A15, TARSKI:def 1;
then A24: dom (SgmX (BagOrder n),sc) = Seg 1 by A17, FINSEQ_1:4, TARSKI:2;
then A25: 1 in dom (SgmX (BagOrder n),sc) by FINSEQ_1:4, TARSKI:def 1;
(SgmX (BagOrder n),sc) /. 1 = (SgmX (BagOrder n),sc) . 1 by A15, PARTFUN1:def 8;
then (SgmX (BagOrder n),sc) /. 1 in rng (SgmX (BagOrder n),sc) by A25, FUNCT_1:12;
then A26: (SgmX (BagOrder n),sc) /. 1 = EmptyBag n by A13, TARSKI:def 1;
A27: len (SgmX (BagOrder n),sc) = 1 by A24, FINSEQ_1:def 3;
dom c = Bags n by FUNCT_2:def 1;
then 1 in dom (c * (SgmX (BagOrder n),sc)) by A13, A15, A16, FUNCT_1:21;
then A28: (c * (SgmX (BagOrder n),sc)) /. 1 = (c * (SgmX (BagOrder n),sc)) . 1 by PARTFUN1:def 8
.= c . ((SgmX (BagOrder n),sc) . 1) by A15, FUNCT_1:23
.= c . (EmptyBag n) by A13, A16, TARSKI:def 1
.= coefficient c by Lm2 ;
dom y = Seg (len y) by FINSEQ_1:def 3
.= dom (SgmX (BagOrder n),sc) by A1, FINSEQ_1:def 3 ;
then y . 1 = y /. 1 by A25, PARTFUN1:def 8
.= ((c * (SgmX (BagOrder n),sc)) /. 1) * (eval (((SgmX (BagOrder n),sc) /. 1) @ ),x) by A1, A3, A27
.= (coefficient c) * (eval (EmptyBag n),x) by A26, A28, POLYNOM2:def 3
.= (coefficient c) * (1. L) by POLYNOM2:16
.= coefficient c by VECTSP_1:def 16 ;
then y = <*(coefficient c)*> by A1, A27, FINSEQ_1:57;
hence eval c,x = coefficient c by A2, RLVECT_1:61; :: thesis: verum

end;

end;

end;

hence eval c,x = coefficient c ; :: thesis: verum