let L be Z_Lattice; :: thesis:
let w be Vector of (DivisibleMod L); :: thesis:
let l1, l2 be Linear_Combination of DivisibleMod L; :: thesis:
set A = ((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2);
set C1 = (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l1);
consider p being FinSequence such that
A1: rng p = (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l1) and
A2: p is one-to-one by FINSEQ_4:58;
reconsider p = p as FinSequence of (DivisibleMod L) by A1, FINSEQ_1:def 4;
A3: ((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2) = (Carrier l1) \/ ((Carrier (l1 + l2)) \/ (Carrier l2)) by XBOOLE_1:4;
set C3 = (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier (l1 + l2));
consider r being FinSequence such that
A4: rng r = (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier (l1 + l2)) and
A5: r is one-to-one by FINSEQ_4:58;
reconsider r = r as FinSequence of (DivisibleMod L) by A4, FINSEQ_1:def 4;
A6: ((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2) = (Carrier (l1 + l2)) \/ ((Carrier l1) \/ (Carrier l2)) by XBOOLE_1:4;
set C2 = (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l2);
consider q being FinSequence such that
A7: rng q = (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l2) and
A8: q is one-to-one by FINSEQ_4:58;
reconsider q = q as FinSequence of (DivisibleMod L) by A7, FINSEQ_1:def 4;
consider F being FinSequence of (DivisibleMod L) such that
A9: F is one-to-one and
A10: rng F = Carrier (l1 + l2) and
A11: SumSc (w,(l1 + l2)) = Sum (ScFS (w,(l1 + l2),F)) by defSumScDM;
set FF = F ^ r;
consider G being FinSequence of (DivisibleMod L) such that
A12: G is one-to-one and
A13: rng G = Carrier l1 and
A14: SumSc (w,l1) = Sum (ScFS (w,l1,G)) by defSumScDM;
rng (F ^ r) = (rng F) \/ (rng r) by FINSEQ_1:31;
then rng (F ^ r) = (Carrier (l1 + l2)) \/ (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) by A10, A4, XBOOLE_1:39;
then A15: rng (F ^ r) = ((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2) by A6, XBOOLE_1:7, XBOOLE_1:12;
set GG = G ^ p;
rng (G ^ p) = (rng G) \/ (rng p) by FINSEQ_1:31;
then rng (G ^ p) = (Carrier l1) \/ (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) by A13, A1, XBOOLE_1:39;
then A16: rng (G ^ p) = ((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2) by A3, XBOOLE_1:7, XBOOLE_1:12;
rng F misses rng r

proof

set x = the Element of (rng F) /\ (rng r);
assume not rng F misses rng r ; :: thesis:
then ( the Element of (rng F) /\ (rng r) in Carrier (l1 + l2) & the Element of (rng F) /\ (rng r) in (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier (l1 + l2)) ) by A10, A4, XBOOLE_0:def 4;
hence contradiction by XBOOLE_0:def 5; :: thesis:

end;

then A17: F ^ r is one-to-one by A9, A5, FINSEQ_3:91;
rng G misses rng p

proof

set x = the Element of (rng G) /\ (rng p);
assume not rng G misses rng p ; :: thesis:
then ( the Element of (rng G) /\ (rng p) in Carrier l1 & the Element of (rng G) /\ (rng p) in (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l1) ) by A13, A1, XBOOLE_0:def 4;
hence contradiction by XBOOLE_0:def 5; :: thesis:

end;

then A18: G ^ p is one-to-one by A12, A2, FINSEQ_3:91;
then A19: len (G ^ p) = len (F ^ r) by A17, A16, A15, FINSEQ_1:48;
deffunc H₁( Nat) -> set = (F ^ r) <- ((G ^ p) . $1);
consider P being FinSequence such that
A20: len P = len (F ^ r) and
A21: for k being Nat st k in dom P holds
P . k = H₁(k) from FINSEQ_1:sch 2();
A22: dom P = Seg (len (F ^ r)) by A20, FINSEQ_1:def 3;

A23: now :: thesis:

let x be object ; :: thesis:
assume A24: x in dom (G ^ p) ; :: thesis:
then reconsider n = x as Element of NAT ;
(G ^ p) . n in rng (F ^ r) by A16, A15, A24, FUNCT_1:def 3;
then A25: F ^ r just_once_values (G ^ p) . n by A17, FINSEQ_4:8;
n in Seg (len (F ^ r)) by A19, A24, FINSEQ_1:def 3;
then (F ^ r) . (P . n) = (F ^ r) . ((F ^ r) <- ((G ^ p) . n)) by A21, A22
.= (G ^ p) . n by A25, FINSEQ_4:def 3 ;
hence (G ^ p) . x = (F ^ r) . (P . x) ; :: thesis:

end;

A26: rng P c= dom (F ^ r)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in rng P ; :: thesis:
then consider y being object such that
A27: y in dom P and
A28: P . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A27;
y in dom (G ^ p) by A19, A20, A27, FINSEQ_3:29;
then (G ^ p) . y in rng (F ^ r) by A16, A15, FUNCT_1:def 3;
then A29: F ^ r just_once_values (G ^ p) . y by A17, FINSEQ_4:8;
P . y = (F ^ r) <- ((G ^ p) . y) by A21, A27;
hence x in dom (F ^ r) by A28, A29, FINSEQ_4:def 3; :: thesis:

end;

now :: thesis:

let x be object ; :: thesis:
thus ( x in dom (G ^ p) implies ( x in dom P & P . x in dom (F ^ r) ) ) :: thesis:

proof

assume x in dom (G ^ p) ; :: thesis:
then x in Seg (len P) by A19, A20, FINSEQ_1:def 3;
hence x in dom P by FINSEQ_1:def 3; :: thesis:
then P . x in rng P by FUNCT_1:def 3;
hence P . x in dom (F ^ r) by A26; :: thesis:

end;

assume that
A30: x in dom P and
P . x in dom (F ^ r) ; :: thesis:
x in Seg (len P) by A30, FINSEQ_1:def 3;
hence x in dom (G ^ p) by A19, A20, FINSEQ_1:def 3; :: thesis:

end;

then A31: G ^ p = (F ^ r) * P by A23, FUNCT_1:10;
dom (F ^ r) c= rng P

proof

set f = ((F ^ r) ") * (G ^ p);
let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A32: x in dom (F ^ r) ; :: thesis:
dom ((F ^ r) ") = rng (G ^ p) by A17, A16, A15, FUNCT_1:33;
then A33: rng (((F ^ r) ") * (G ^ p)) = rng ((F ^ r) ") by RELAT_1:28
.= dom (F ^ r) by A17, FUNCT_1:33 ;
((F ^ r) ") * (G ^ p) = (((F ^ r) ") * (F ^ r)) * P by A31, RELAT_1:36
.= (id (dom (F ^ r))) * P by A17, FUNCT_1:39
.= P by A26, RELAT_1:53 ;
hence x in rng P by A32, A33; :: thesis:

end;

then A34: dom (F ^ r) = rng P by A26;
A35: len r = len (ScFS (w,(l1 + l2),r)) by defScFSDM;
B1: dom r = Seg (len r) by FINSEQ_1:def 3
.= Seg (len (ScFS (w,(l1 + l2),r))) by defScFSDM
.= dom (ScFS (w,(l1 + l2),r)) by FINSEQ_1:def 3 ;

now :: thesis:

let k be Nat; :: thesis:
assume A36: k in dom (ScFS (w,(l1 + l2),r)) ; :: thesis:
then r /. k = r . k by B1, PARTFUN1:def 6;
then r /. k in (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier (l1 + l2)) by A4, A36, B1, FUNCT_1:def 3;
then A37: not r /. k in Carrier (l1 + l2) by XBOOLE_0:def 5;
(ScFS (w,(l1 + l2),r)) . k = (ScProductDM L) . (w,(((l1 + l2) . (r /. k)) * (r /. k))) by A36, defScFSDM;
then (ScFS (w,(l1 + l2),r)) . k = (ScProductDM L) . (w,((0. INT.Ring) * (r /. k))) by A37
.= (ScProductDM L) . (w,(0. (DivisibleMod L))) by VECTSP_1:14
.= 0. F_Real by ZMODLAT2:14 ;
hence (ScFS (w,(l1 + l2),r)) . k = - ((ScFS (w,(l1 + l2),r)) . k)
.= - ((ScFS (w,(l1 + l2),r)) /. k) by A36, PARTFUN1:def 6 ;
:: thesis:

end;

then A38: Sum (ScFS (w,(l1 + l2),r)) = - (Sum (ScFS (w,(l1 + l2),r))) by A35, RLVECT_2:4;
A39: len p = len (ScFS (w,l1,p)) by defScFSDM;
B1: dom p = Seg (len p) by FINSEQ_1:def 3
.= Seg (len (ScFS (w,l1,p))) by defScFSDM
.= dom (ScFS (w,l1,p)) by FINSEQ_1:def 3 ;

now :: thesis:

let k be Nat; :: thesis:
assume A40: k in dom (ScFS (w,l1,p)) ; :: thesis:
then p /. k = p . k by B1, PARTFUN1:def 6;
then p /. k in (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l1) by A1, A40, B1, FUNCT_1:def 3;
then A41: not p /. k in Carrier l1 by XBOOLE_0:def 5;
(ScFS (w,l1,p)) . k = (ScProductDM L) . (w,((l1 . (p /. k)) * (p /. k))) by A40, defScFSDM;
then (ScFS (w,l1,p)) . k = (ScProductDM L) . (w,((0. INT.Ring) * (p /. k))) by A41
.= (ScProductDM L) . (w,(0. (DivisibleMod L))) by VECTSP_1:14
.= 0. F_Real by ZMODLAT2:14 ;
hence (ScFS (w,l1,p)) . k = - ((ScFS (w,l1,p)) . k)
.= - ((ScFS (w,l1,p)) /. k) by A40, PARTFUN1:def 6 ;
:: thesis:

end;

then A42: Sum (ScFS (w,l1,p)) = - (Sum (ScFS (w,l1,p))) by A39, RLVECT_2:4;
A43: len q = len (ScFS (w,l2,q)) by defScFSDM;
then B1: dom q = dom (ScFS (w,l2,q)) by FINSEQ_3:29;

now :: thesis:

let k be Nat; :: thesis:
assume A44: k in dom (ScFS (w,l2,q)) ; :: thesis:
then q /. k = q . k by B1, PARTFUN1:def 6;
then q /. k in (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l2) by A7, A44, B1, FUNCT_1:def 3;
then A45: not q /. k in Carrier l2 by XBOOLE_0:def 5;
(ScFS (w,l2,q)) . k = (ScProductDM L) . (w,((l2 . (q /. k)) * (q /. k))) by A44, defScFSDM;
then (ScFS (w,l2,q)) . k = (ScProductDM L) . (w,((0. INT.Ring) * (q /. k))) by A45
.= (ScProductDM L) . (w,(0. (DivisibleMod L))) by VECTSP_1:14
.= 0. F_Real by ZMODLAT2:14 ;
hence (ScFS (w,l2,q)) . k = - ((ScFS (w,l2,q)) . k)
.= - ((ScFS (w,l2,q)) /. k) by A44, PARTFUN1:def 6 ;
:: thesis:

end;

then A46: Sum (ScFS (w,l2,q)) = - (Sum (ScFS (w,l2,q))) by A43, RLVECT_2:4;
set g = ScFS (w,l1,(G ^ p));
A47: len (ScFS (w,l1,(G ^ p))) = len (G ^ p) by defScFSDM;
then A48: Seg (len (G ^ p)) = dom (ScFS (w,l1,(G ^ p))) by FINSEQ_1:def 3;
set f = ScFS (w,(l1 + l2),(F ^ r));
consider H being FinSequence of (DivisibleMod L) such that
A49: H is one-to-one and
A50: rng H = Carrier l2 and
A51: Sum (ScFS (w,l2,H)) = SumSc (w,l2) by defSumScDM;
set HH = H ^ q;
rng H misses rng q

proof

set x = the Element of (rng H) /\ (rng q);
assume not rng H misses rng q ; :: thesis:
then ( the Element of (rng H) /\ (rng q) in Carrier l2 & the Element of (rng H) /\ (rng q) in (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) \ (Carrier l2) ) by A50, A7, XBOOLE_0:def 4;
hence contradiction by XBOOLE_0:def 5; :: thesis:

end;

then A52: H ^ q is one-to-one by A49, A8, FINSEQ_3:91;
rng (H ^ q) = (rng H) \/ (rng q) by FINSEQ_1:31;
then rng (H ^ q) = (Carrier l2) \/ (((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2)) by A50, A7, XBOOLE_1:39;
then A53: rng (H ^ q) = ((Carrier (l1 + l2)) \/ (Carrier l1)) \/ (Carrier l2) by XBOOLE_1:7, XBOOLE_1:12;
then A54: len (G ^ p) = len (H ^ q) by A52, A18, A16, FINSEQ_1:48;
then B1: dom (G ^ p) = dom (H ^ q) by FINSEQ_3:29;
deffunc H₂( Nat) -> set = (H ^ q) <- ((G ^ p) . $1);
consider R being FinSequence such that
A55: len R = len (H ^ q) and
A56: for k being Nat st k in dom R holds
R . k = H₂(k) from FINSEQ_1:sch 2();
A57: dom R = dom (H ^ q) by A55, FINSEQ_3:29;

A58: now :: thesis:

let x be object ; :: thesis:
assume A59: x in dom (G ^ p) ; :: thesis:
then reconsider n = x as Element of NAT ;
(G ^ p) . n in rng (H ^ q) by A16, A53, A59, FUNCT_1:def 3;
then A60: H ^ q just_once_values (G ^ p) . n by A52, FINSEQ_4:8;
(H ^ q) . (R . n) = (H ^ q) . ((H ^ q) <- ((G ^ p) . n)) by A56, A57, B1, A59
.= (G ^ p) . n by A60, FINSEQ_4:def 3 ;
hence (G ^ p) . x = (H ^ q) . (R . x) ; :: thesis:

end;

A61: rng R c= dom (H ^ q)

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in rng R ; :: thesis:
then consider y being object such that
A62: y in dom R and
A63: R . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A62;
y in dom (G ^ p) by A54, A55, A62, FINSEQ_3:29;
then (G ^ p) . y in rng (H ^ q) by A16, A53, FUNCT_1:def 3;
then A64: H ^ q just_once_values (G ^ p) . y by A52, FINSEQ_4:8;
R . y = (H ^ q) <- ((G ^ p) . y) by A56, A62;
hence x in dom (H ^ q) by A63, A64, FINSEQ_4:def 3; :: thesis:

end;

now :: thesis:

let x be object ; :: thesis:
thus ( x in dom (G ^ p) implies ( x in dom R & R . x in dom (H ^ q) ) ) :: thesis:

proof

assume x in dom (G ^ p) ; :: thesis:
hence x in dom R by A55, FINSEQ_3:29, A54; :: thesis:
then R . x in rng R by FUNCT_1:def 3;
hence R . x in dom (H ^ q) by A61; :: thesis:

end;

assume ( x in dom R & R . x in dom (H ^ q) ) ; :: thesis:
hence x in dom (G ^ p) by A54, A55, FINSEQ_3:29; :: thesis:

end;

then A66: G ^ p = (H ^ q) * R by A58, FUNCT_1:10;
dom (H ^ q) c= rng R

proof

set f = ((H ^ q) ") * (G ^ p);
let x be object ; :: according to TARSKI:def 3 :: thesis:
assume A67: x in dom (H ^ q) ; :: thesis:
dom ((H ^ q) ") = rng (G ^ p) by A52, A16, A53, FUNCT_1:33;
then A68: rng (((H ^ q) ") * (G ^ p)) = rng ((H ^ q) ") by RELAT_1:28
.= dom (H ^ q) by A52, FUNCT_1:33 ;
((H ^ q) ") * (G ^ p) = (((H ^ q) ") * (H ^ q)) * R by A66, RELAT_1:36
.= (id (dom (H ^ q))) * R by A52, FUNCT_1:39
.= R by A61, RELAT_1:53 ;
hence x in rng R by A67, A68; :: thesis:

end;

then A69: dom (H ^ q) = rng R by A61;
set h = ScFS (w,l2,(H ^ q));
A70: Sum (ScFS (w,l2,(H ^ q))) = Sum ((ScFS (w,l2,H)) ^ (ScFS (w,l2,q))) by ThScFSDM6
.= (Sum (ScFS (w,l2,H))) + (0. F_Real) by A46, RLVECT_1:41
.= Sum (ScFS (w,l2,H)) ;
A71: Sum (ScFS (w,l1,(G ^ p))) = Sum ((ScFS (w,l1,G)) ^ (ScFS (w,l1,p))) by ThScFSDM6
.= (Sum (ScFS (w,l1,G))) + (0. F_Real) by A42, RLVECT_1:41
.= Sum (ScFS (w,l1,G)) ;
A72: dom P = dom (F ^ r) by A20, FINSEQ_3:29;
A73: P is one-to-one by A20, A34, FINSEQ_3:29, FINSEQ_4:60;
A74: dom R = dom (H ^ q) by A55, FINSEQ_3:29;
A75: R is one-to-one by A55, A69, FINSEQ_3:29, FINSEQ_4:60;
R is Function of (dom (H ^ q)),(dom (H ^ q)) by A61, A74, FUNCT_2:2;
then reconsider R = R as Permutation of (dom (H ^ q)) by A69, A75, FUNCT_2:57;
A76: len (ScFS (w,l2,(H ^ q))) = len (H ^ q) by defScFSDM;
then dom (ScFS (w,l2,(H ^ q))) = dom (H ^ q) by FINSEQ_3:29;
then reconsider R = R as Permutation of (dom (ScFS (w,l2,(H ^ q)))) ;
reconsider Hp = (ScFS (w,l2,(H ^ q))) * R as FinSequence of F_Real by FINSEQ_2:47;
A77: len Hp = len (G ^ p) by A54, A76, FINSEQ_2:44;
reconsider P = P as Function of (dom (F ^ r)),(dom (F ^ r)) by A26, A72, FUNCT_2:2;
reconsider P = P as Permutation of (dom (F ^ r)) by A34, A73, FUNCT_2:57;
A78: len (ScFS (w,(l1 + l2),(F ^ r))) = len (F ^ r) by defScFSDM;
then dom (ScFS (w,(l1 + l2),(F ^ r))) = dom (F ^ r) by FINSEQ_3:29;
then reconsider P = P as Permutation of (dom (ScFS (w,(l1 + l2),(F ^ r)))) ;
reconsider Fp = (ScFS (w,(l1 + l2),(F ^ r))) * P as FinSequence of F_Real by FINSEQ_2:47;
A79: Sum (ScFS (w,(l1 + l2),(F ^ r))) = Sum ((ScFS (w,(l1 + l2),F)) ^ (ScFS (w,(l1 + l2),r))) by ThScFSDM6
.= (Sum (ScFS (w,(l1 + l2),F))) + (0. F_Real) by A38, RLVECT_1:41
.= Sum (ScFS (w,(l1 + l2),F)) ;
deffunc H₃( Nat) -> Element of the carrier of F_Real = ((ScFS (w,l1,(G ^ p))) /. $1) + (Hp /. $1);
consider I being FinSequence such that
A80: len I = len (G ^ p) and
A81: for k being Nat st k in dom I holds
I . k = H₃(k) from FINSEQ_1:sch 2();
A83: dom I = Seg (len (G ^ p)) by A80, FINSEQ_1:def 3;
rng I c= the carrier of F_Real

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in rng I ; :: thesis:
then consider y being object such that
A84: y in dom I and
A85: I . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A84;
I . y = ((ScFS (w,l1,(G ^ p))) /. y) + (Hp /. y) by A81, A84;
hence x in the carrier of F_Real by A85; :: thesis:

end;

then reconsider I = I as FinSequence of F_Real by FINSEQ_1:def 4;
A86: len Fp = len I by A19, A78, A80, FINSEQ_2:44;
then X1: ( dom I = Seg (len I) & dom Fp = Seg (len I) ) by FINSEQ_1:def 3;
X2: dom Hp = Seg (len Hp) by FINSEQ_1:def 3
.= dom I by A54, A76, A80, FINSEQ_1:def 3, FINSEQ_2:44 ;
X3: dom (H ^ q) = Seg (len (H ^ q)) by FINSEQ_1:def 3
.= dom I by A18, A16, A52, A53, A80, FINSEQ_1:48, FINSEQ_1:def 3 ;
then X4: dom (G ^ p) = dom I by A54, FINSEQ_3:29;
X5: dom (ScFS (w,l1,(G ^ p))) = Seg (len (ScFS (w,l1,(G ^ p)))) by FINSEQ_1:def 3
.= dom I by A80, defScFSDM, FINSEQ_1:def 3 ;

A87: now :: thesis:

let k be Nat; :: thesis:
assume A88: k in dom I ; :: thesis:
then A90: Hp /. k = ((ScFS (w,l2,(H ^ q))) * R) . k by X2, PARTFUN1:def 6
.= (ScFS (w,l2,(H ^ q))) . (R . k) by A88, X2, FUNCT_1:12 ;
set v = (G ^ p) /. k;
A91: Fp . k = (ScFS (w,(l1 + l2),(F ^ r))) . (P . k) by A88, X1, FUNCT_1:12;
R . k in dom R by A69, A74, A88, X3, FUNCT_1:def 3;
then reconsider j = R . k as Element of NAT by FINSEQ_3:23;
A94: (H ^ q) . j = (G ^ p) . k by A58, A88, X3, B1
.= (G ^ p) /. k by A88, X3, B1, PARTFUN1:def 6 ;
A95: dom (F ^ r) = dom (G ^ p) by A19, FINSEQ_3:29;
then P . k in dom P by A34, A72, A88, X3, FUNCT_1:def 3, B1;
then reconsider l = P . k as Element of NAT by FINSEQ_3:23;
A96: (F ^ r) . l = (G ^ p) . k by A23, A88, X3, B1
.= (G ^ p) /. k by A88, X3, B1, PARTFUN1:def 6 ;
R . k in dom (H ^ q) by A69, A74, A88, X3, FUNCT_1:def 3;
then A97: (ScFS (w,l2,(H ^ q))) . j = (ScProductDM L) . (w,((l2 . ((G ^ p) /. k)) * ((G ^ p) /. k))) by A94, ThScFSDM1;
P . k in dom (F ^ r) by A34, A72, A88, A95, X4, FUNCT_1:def 3;
then A98: (ScFS (w,(l1 + l2),(F ^ r))) . l = (ScProductDM L) . (w,(((l1 + l2) . ((G ^ p) /. k)) * ((G ^ p) /. k))) by A96, ThScFSDM1
.= (ScProductDM L) . (w,(((l1 . ((G ^ p) /. k)) + (l2 . ((G ^ p) /. k))) * ((G ^ p) /. k))) by VECTSP_6:22
.= (ScProductDM L) . (w,(((l1 . ((G ^ p) /. k)) * ((G ^ p) /. k)) + ((l2 . ((G ^ p) /. k)) * ((G ^ p) /. k)))) by VECTSP_1:def 15
.= ((ScProductDM L) . (w,((l1 . ((G ^ p) /. k)) * ((G ^ p) /. k)))) + ((ScProductDM L) . (w,((l2 . ((G ^ p) /. k)) * ((G ^ p) /. k)))) by ZMODLAT2:12 ;
(ScFS (w,l1,(G ^ p))) /. k = (ScFS (w,l1,(G ^ p))) . k by A88, X5, PARTFUN1:def 6
.= (ScProductDM L) . (w,((l1 . ((G ^ p) /. k)) * ((G ^ p) /. k))) by A88, X5, defScFSDM ;
hence I . k = Fp . k by A81, A88, A90, A97, A91, A98; :: thesis:

end;

( dom I = Seg (len I) & dom Fp = Seg (len I) ) by A86, FINSEQ_1:def 3;
then A100: I = Fp by A87;
( Sum Fp = Sum (ScFS (w,(l1 + l2),(F ^ r))) & Sum Hp = Sum (ScFS (w,l2,(H ^ q))) ) by RLVECT_2:7;
hence SumSc (w,(l1 + l2)) = (SumSc (w,l1)) + (SumSc (w,l2)) by A11, A14, A47, A48, A51, A71, A70, A77, A79, A80, A81, A83, A100, RLVECT_2:2; :: thesis: