let a, c be Element of L; ( ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & a = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x)) ) ) & ex y being FinSequence of the carrier of L st
( len y = len (SgmX ((BagOrder n),(Support p))) & c = Sum y & ( for i being Element of NAT st 1 <= i & i <= len y holds
y /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x)) ) ) implies a = c )
assume that
A4:
ex y1 being FinSequence of the carrier of L st
( len y1 = len (SgmX ((BagOrder n),(Support p))) & a = Sum y1 & ( for i being Element of NAT st 1 <= i & i <= len y1 holds
y1 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x)) ) )
and
A5:
ex y2 being FinSequence of the carrier of L st
( len y2 = len (SgmX ((BagOrder n),(Support p))) & c = Sum y2 & ( for i being Element of NAT st 1 <= i & i <= len y2 holds
y2 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x)) ) )
; a = c
consider y1 being FinSequence of the carrier of L such that
A6:
len y1 = len (SgmX ((BagOrder n),(Support p)))
and
A7:
a = Sum y1
and
A8:
for i being Element of NAT st 1 <= i & i <= len y1 holds
y1 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x))
by A4;
consider y2 being FinSequence of the carrier of L such that
A9:
len y2 = len (SgmX ((BagOrder n),(Support p)))
and
A10:
c = Sum y2
and
A11:
for i being Element of NAT st 1 <= i & i <= len y2 holds
y2 /. i = ((p * (SgmX ((BagOrder n),(Support p)))) /. i) * (eval (((SgmX ((BagOrder n),(Support p))) /. i),x))
by A5;
for k being Nat st 1 <= k & k <= len y1 holds
y1 . k = y2 . k
proof
let k be
Nat;
( 1 <= k & k <= len y1 implies y1 . k = y2 . k )
assume that A12:
1
<= k
and A13:
k <= len y1
;
y1 . k = y2 . k
k in Seg (len y2)
by A6, A9, A12, A13, FINSEQ_1:1;
then A14:
k in dom y2
by FINSEQ_1:def 3;
A15:
k in Seg (len y1)
by A12, A13, FINSEQ_1:1;
then
k in dom y1
by FINSEQ_1:def 3;
hence y1 . k =
y1 /. k
by PARTFUN1:def 6
.=
((p * (SgmX ((BagOrder n),(Support p)))) /. k) * (eval (((SgmX ((BagOrder n),(Support p))) /. k),x))
by A8, A12, A13, A15
.=
y2 /. k
by A6, A9, A11, A12, A13, A15
.=
y2 . k
by A14, PARTFUN1:def 6
;
verum
end;
hence
a = c
by A6, A7, A9, A10, FINSEQ_1:14; verum