let n be Ordinal; :: thesis: for L being non empty non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr
for x being Function of n,L holds eval (1_ n,L),x = 1. L
let L be non empty non trivial right_complementable add-associative right_zeroed well-unital distributive doubleLoopStr ; :: thesis: for x being Function of n,L holds eval (1_ n,L),x = 1. L
let x be Function of n,L; :: thesis: eval (1_ n,L),x = 1. L
set 1p = 1_ n,L;
A1: for u being set st u in {(EmptyBag n)} holds
u in Support (1_ n,L)

proof

let u be set ; :: thesis: ( u in {(EmptyBag n)} implies u in Support (1_ n,L) )
assume A2: u in {(EmptyBag n)} ; :: thesis: u in Support (1_ n,L)
then A3: u = EmptyBag n by TARSKI:def 1;
reconsider u = u as Element of Bags n by A2, TARSKI:def 1;
(1_ n,L) . u = 1. L by A3, POLYNOM1:84;
then (1_ n,L) . u <> 0. L by VECTSP_1:def 13, VECTSP_1:36;
hence u in Support (1_ n,L) by POLYNOM1:def 9; :: thesis: verum

end;

reconsider s1p = Support (1_ n,L) as finite Subset of (Bags n) ;
set sg1p = SgmX (BagOrder n),s1p;
A4: BagOrder n linearly_orders Support (1_ n,L) by Th20;
for u being set st u in Support (1_ n,L) holds
u in {(EmptyBag n)}

proof

let u be set ; :: thesis: ( u in Support (1_ n,L) implies u in {(EmptyBag n)} )
assume A5: u in Support (1_ n,L) ; :: thesis: u in {(EmptyBag n)}
then reconsider u = u as Element of Bags n ;
(1_ n,L) . u <> 0. L by A5, POLYNOM1:def 9;
then u = EmptyBag n by POLYNOM1:84;
hence u in {(EmptyBag n)} by TARSKI:def 1; :: thesis: verum

end;

then A6: Support (1_ n,L) = {(EmptyBag n)} by A1, TARSKI:2;
then A7: rng (SgmX (BagOrder n),s1p) = {(EmptyBag n)} by A4, PRE_POLY:def 2;
then A8: EmptyBag n in rng (SgmX (BagOrder n),s1p) by TARSKI:def 1;
then A9: 1 in dom (SgmX (BagOrder n),s1p) by FINSEQ_3:33;
then (SgmX (BagOrder n),s1p) /. 1 = (SgmX (BagOrder n),s1p) . 1 by PARTFUN1:def 8;
then (SgmX (BagOrder n),s1p) /. 1 in rng (SgmX (BagOrder n),s1p) by A9, FUNCT_1:def 5;
then A10: (SgmX (BagOrder n),s1p) /. 1 = EmptyBag n by A7, TARSKI:def 1;
A11: for u being set st u in dom (SgmX (BagOrder n),s1p) holds
u in {1}

proof

let u be set ; :: thesis: ( u in dom (SgmX (BagOrder n),s1p) implies u in {1} )
assume A12: u in dom (SgmX (BagOrder n),s1p) ; :: thesis: u in {1}
assume A13: not u in {1} ; :: thesis: contradiction
reconsider u = u as Element of NAT by A12;
(SgmX (BagOrder n),s1p) /. u = (SgmX (BagOrder n),s1p) . u by A12, PARTFUN1:def 8;
then A14: (SgmX (BagOrder n),s1p) /. u in rng (SgmX (BagOrder n),s1p) by A12, FUNCT_1:def 5;
A15: u <> 1 by A13, TARSKI:def 1;
A16: 1 < u

proof

consider k being Nat such that
A17: dom (SgmX (BagOrder n),s1p) = 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 A12, A17;
hence 1 < u by A15, XXREAL_0:1; :: thesis: verum

end;

(SgmX (BagOrder n),s1p) /. 1 = (SgmX (BagOrder n),s1p) . 1 by A8, A12, FINSEQ_3:33, PARTFUN1:def 8;
then (SgmX (BagOrder n),s1p) /. 1 in rng (SgmX (BagOrder n),s1p) by A9, FUNCT_1:def 5;
then (SgmX (BagOrder n),s1p) /. 1 = EmptyBag n by A7, TARSKI:def 1
.= (SgmX (BagOrder n),s1p) /. u by A7, A14, TARSKI:def 1 ;
hence contradiction by A4, A9, A12, A16, PRE_POLY:def 2; :: thesis: verum

end;

A18: dom (1_ n,L) = Bags n by FUNCT_2:def 1;
A19: 1 in dom (SgmX (BagOrder n),s1p) by A8, FINSEQ_3:33;
A20: (SgmX (BagOrder n),s1p) . 1 in rng (SgmX (BagOrder n),s1p) by A9, FUNCT_1:def 5;
then (SgmX (BagOrder n),s1p) . 1 in {(EmptyBag n)} by A6, A4, PRE_POLY:def 2;
then (SgmX (BagOrder n),s1p) . 1 = EmptyBag n by TARSKI:def 1;
then 1 in dom ((1_ n,L) * (SgmX (BagOrder n),s1p)) by A19, A18, FUNCT_1:21;
then A21: ((1_ n,L) * (SgmX (BagOrder n),s1p)) /. 1 = ((1_ n,L) * (SgmX (BagOrder n),s1p)) . 1 by PARTFUN1:def 8
.= (1_ n,L) . ((SgmX (BagOrder n),s1p) . 1) by A9, FUNCT_1:23
.= (1_ n,L) . (EmptyBag n) by A7, A20, TARSKI:def 1
.= 1. L by POLYNOM1:84 ;
consider y being FinSequence of the carrier of L such that
A22: len y = len (SgmX (BagOrder n),s1p) and
A23: Sum y = eval (1_ n,L),x and
A24: for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = (((1_ n,L) * (SgmX (BagOrder n),s1p)) /. i) * (eval (((SgmX (BagOrder n),s1p) /. i) @ ),x) by Def4;
for u being set st u in {1} holds
u in dom (SgmX (BagOrder n),s1p) by A9, TARSKI:def 1;
then dom (SgmX (BagOrder n),s1p) = Seg 1 by A11, FINSEQ_1:4, TARSKI:2;
then A25: len (SgmX (BagOrder n),s1p) = 1 by FINSEQ_1:def 3;
then y . 1 = y /. 1 by A22, FINSEQ_4:24
.= (((1_ n,L) * (SgmX (BagOrder n),s1p)) /. 1) * (eval (((SgmX (BagOrder n),s1p) /. 1) @ ),x) by A25, A22, A24
.= (((1_ n,L) * (SgmX (BagOrder n),s1p)) /. 1) * (1. L) by A10, Th16
.= ((1_ n,L) * (SgmX (BagOrder n),s1p)) /. 1 by VECTSP_1:def 16 ;
then y = <*(1. L)*> by A25, A22, A21, FINSEQ_1:57;
hence eval (1_ n,L),x = 1. L by A23, RLVECT_1:61; :: thesis: verum