let a, c be Element of L; :: thesis: ( 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)) ) ) ; :: thesis: 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; :: thesis: ( 1 <= k & k <= len y1 implies y1 . k = y2 . k )
assume that
A12: 1 <= k and
A13: k <= len y1 ; :: thesis: 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 ;
:: thesis: verum

end;

hence a = c by A6, A7, A9, A10, FINSEQ_1:14; :: thesis: verum