let V be Z_Module; :: thesis:
let L1, L2 be Z_Linear_Combination of V; :: thesis:
consider F being FinSequence of the carrier of V such that
A1: F is one-to-one and
A2: rng F = Carrier (L1 + L2) and
A3: Sum ((L1 + L2) (#) F) = Sum (L1 + L2) by Def23;
set A = ((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2);
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 the carrier of V by A4, FINSEQ_1:def 4;
set FF = F ^ r;
A6: ((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2) = (Carrier (L1 + L2)) \/ ((Carrier L1) \/ (Carrier L2)) by XBOOLE_1:4;
rng F misses rng r

proof

set x = the Element of (rng F) /\ (rng r);
assume not rng F misses rng r ; :: thesis:
then (rng F) /\ (rng r) <> {} ;
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 A2, A4, XBOOLE_0:def 4;
hence contradiction by XBOOLE_0:def 5; :: thesis:

end;

then A7: F ^ r is one-to-one by A1, A5, FINSEQ_3:91;
A8: len r = len ((L1 + L2) (#) r) by Def22;

now :: thesis:

let k be Nat; :: thesis:
assume A9: k in dom r ; :: thesis:
then r /. k = r . k by PARTFUN1:def 6;
then r /. k in (((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2)) \ (Carrier (L1 + L2)) by A4, A9, FUNCT_1:def 3;
then A10: not r /. k in Carrier (L1 + L2) by XBOOLE_0:def 5;
k in dom ((L1 + L2) (#) r) by A8, A9, FINSEQ_3:29;
then ((L1 + L2) (#) r) . k = ((L1 + L2) . (r /. k)) * (r /. k) by Def22;
hence ((L1 + L2) (#) r) . k = 0 * (r /. k) by A10; :: thesis:

end;

then A11: Sum ((L1 + L2) (#) r) = 0 * (Sum r) by A8, ZMODUL01:17
.= 0. V by ZMODUL01:1 ;
set f = (L1 + L2) (#) (F ^ r);
set C1 = (((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2)) \ (Carrier L1);
consider G being FinSequence of the carrier of V such that
A12: G is one-to-one and
A13: rng G = Carrier L1 and
A14: Sum (L1 (#) G) = Sum L1 by Def23;
consider p being FinSequence such that
A15: rng p = (((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2)) \ (Carrier L1) and
A16: p is one-to-one by FINSEQ_4:58;
reconsider p = p as FinSequence of the carrier of V by A15, FINSEQ_1:def 4;
set GG = G ^ p;
A17: Sum ((L1 + L2) (#) (F ^ r)) = Sum (((L1 + L2) (#) F) ^ ((L1 + L2) (#) r)) by Lm2
.= (Sum ((L1 + L2) (#) F)) + (0. V) by A11, RLVECT_1:41
.= Sum ((L1 + L2) (#) F) by RLVECT_1:4 ;
set C2 = (((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2)) \ (Carrier L2);
consider H being FinSequence of the carrier of V such that
A18: H is one-to-one and
A19: rng H = Carrier L2 and
A20: Sum (L2 (#) H) = Sum L2 by Def23;
consider q being FinSequence such that
A21: rng q = (((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2)) \ (Carrier L2) and
A22: q is one-to-one by FINSEQ_4:58;
reconsider q = q as FinSequence of the carrier of V by A21, FINSEQ_1:def 4;
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 (rng H) /\ (rng q) <> {} ;
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 A19, A21, XBOOLE_0:def 4;
hence contradiction by XBOOLE_0:def 5; :: thesis:

end;

then A23: H ^ q is one-to-one by A18, A22, FINSEQ_3:91;
set h = L2 (#) (H ^ q);
A24: ((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2) = (Carrier L1) \/ ((Carrier (L1 + L2)) \/ (Carrier L2)) by XBOOLE_1:4;
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, A15, XBOOLE_1:39;
then A25: rng (G ^ p) = ((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2) by A24, XBOOLE_1:7, XBOOLE_1:12;
A26: len q = len (L2 (#) q) by Def22;

now :: thesis:

let k be Nat; :: thesis:
assume A27: k in dom q ; :: thesis:
then q /. k = q . k by PARTFUN1:def 6;
then q /. k in (((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2)) \ (Carrier L2) by A21, A27, FUNCT_1:def 3;
then A28: not q /. k in Carrier L2 by XBOOLE_0:def 5;
k in dom (L2 (#) q) by A26, A27, FINSEQ_3:29;
then (L2 (#) q) . k = (L2 . (q /. k)) * (q /. k) by Def22;
hence (L2 (#) q) . k = 0 * (q /. k) by A28; :: thesis:

end;

then A29: Sum (L2 (#) q) = 0 * (Sum q) by A26, ZMODUL01:17
.= 0. V by ZMODUL01:1 ;
A30: Sum (L2 (#) (H ^ q)) = Sum ((L2 (#) H) ^ (L2 (#) q)) by Lm2
.= (Sum (L2 (#) H)) + (0. V) by A29, RLVECT_1:41
.= Sum (L2 (#) H) by RLVECT_1:4 ;
deffunc H₁( Nat) -> set = (F ^ r) <- ((G ^ p) . $1);
set g = L1 (#) (G ^ p);
consider P being FinSequence such that
A31: len P = len (F ^ r) and
A32: for k being Nat st k in dom P holds
P . k = H₁(k) from FINSEQ_1:sch 2();
A33: dom P = Seg (len (F ^ r)) by A31, FINSEQ_1:def 3;
A34: len p = len (L1 (#) p) by Def22;

now :: thesis:

let k be Nat; :: thesis:
assume A35: k in dom p ; :: thesis:
then p /. k = p . k by PARTFUN1:def 6;
then p /. k in (((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2)) \ (Carrier L1) by A15, A35, FUNCT_1:def 3;
then A36: not p /. k in Carrier L1 by XBOOLE_0:def 5;
k in dom (L1 (#) p) by A34, A35, FINSEQ_3:29;
then (L1 (#) p) . k = (L1 . (p /. k)) * (p /. k) by Def22;
hence (L1 (#) p) . k = 0 * (p /. k) by A36; :: thesis:

end;

then A37: Sum (L1 (#) p) = 0 * (Sum p) by A34, ZMODUL01:17
.= 0. V by ZMODUL01:1 ;
A38: Sum (L1 (#) (G ^ p)) = Sum ((L1 (#) G) ^ (L1 (#) p)) by Lm2
.= (Sum (L1 (#) G)) + (0. V) by A37, RLVECT_1:41
.= Sum (L1 (#) G) by RLVECT_1:4 ;
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 A2, A4, XBOOLE_1:39;
then A39: rng (F ^ r) = ((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2) by A6, XBOOLE_1:7, XBOOLE_1:12;
rng G misses rng p

proof

set x = the Element of (rng G) /\ (rng p);
assume not rng G misses rng p ; :: thesis:
then (rng G) /\ (rng p) <> {} ;
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, A15, XBOOLE_0:def 4;
hence contradiction by XBOOLE_0:def 5; :: thesis:

end;

then A40: G ^ p is one-to-one by A12, A16, FINSEQ_3:91;
then A41: len (G ^ p) = len (F ^ r) by A7, A25, A39, FINSEQ_1:48;
A42: dom P = Seg (len (F ^ r)) by A31, FINSEQ_1:def 3;

A43: now :: thesis:

let x be object ; :: thesis:
assume A44: x in dom (G ^ p) ; :: thesis:
then reconsider n = x as Element of NAT ;
(G ^ p) . n in rng (F ^ r) by A25, A39, A44, FUNCT_1:def 3;
then A45: F ^ r just_once_values (G ^ p) . n by A7, FINSEQ_4:8;
n in Seg (len (F ^ r)) by A41, A44, FINSEQ_1:def 3;
then (F ^ r) . (P . n) = (F ^ r) . ((F ^ r) <- ((G ^ p) . n)) by A32, A42
.= (G ^ p) . n by A45, FINSEQ_4:def 3 ;
hence (G ^ p) . x = (F ^ r) . (P . x) ; :: thesis:

end;

A46: rng P c= Seg (len (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
A47: y in dom P and
A48: P . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A47;
y in Seg (len (F ^ r)) by A31, A47, FINSEQ_1:def 3;
then y in dom (G ^ p) by A41, FINSEQ_1:def 3;
then (G ^ p) . y in rng (F ^ r) by A25, A39, FUNCT_1:def 3;
then A49: F ^ r just_once_values (G ^ p) . y by A7, FINSEQ_4:8;
P . y = (F ^ r) <- ((G ^ p) . y) by A32, A47;
then P . y in dom (F ^ r) by A49, FINSEQ_4:def 3;
hence x in Seg (len (F ^ r)) by A48, FINSEQ_1: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 A41, A31, 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;
then P . x in Seg (len (F ^ r)) by A46;
hence P . x in dom (F ^ r) by FINSEQ_1:def 3; :: thesis:

end;

assume that
A50: x in dom P and
P . x in dom (F ^ r) ; :: thesis:
x in Seg (len P) by A50, FINSEQ_1:def 3;
hence x in dom (G ^ p) by A41, A31, FINSEQ_1:def 3; :: thesis:

end;

then A51: G ^ p = (F ^ r) * P by A43, FUNCT_1:10;
Seg (len (F ^ r)) c= rng P

proof

set f = ((F ^ r) ") * (G ^ p);

now :: thesis:

let x be object ; :: thesis:
assume A52: x in Seg (len (F ^ r)) ; :: thesis:
dom ((F ^ r) ") = rng (G ^ p) by A7, A25, A39, FUNCT_1:33;
then A53: rng (((F ^ r) ") * (G ^ p)) = rng ((F ^ r) ") by RELAT_1:28
.= dom (F ^ r) by A7, FUNCT_1:33 ;
A54: rng P c= dom (F ^ r) by A46, FINSEQ_1:def 3;
((F ^ r) ") * (G ^ p) = (((F ^ r) ") * (F ^ r)) * P by A51, RELAT_1:36
.= (id (dom (F ^ r))) * P by A7, FUNCT_1:39
.= P by A54, RELAT_1:53 ;
hence x in rng P by A52, A53, FINSEQ_1:def 3; :: thesis:

end;

hence Seg (len (F ^ r)) c= rng P ; :: thesis:

end;

then A55: Seg (len (F ^ r)) = rng P by A46;
then A56: P is one-to-one by A33, FINSEQ_4:60;
reconsider P = P as Function of (Seg (len (F ^ r))),(Seg (len (F ^ r))) by FUNCT_2:1, A55, A42;
reconsider P = P as Permutation of (Seg (len (F ^ r))) by A55, A56, FUNCT_2:57;
A57: len ((L1 + L2) (#) (F ^ r)) = len (F ^ r) by Def22;
then A58: Seg (len (F ^ r)) = dom ((L1 + L2) (#) (F ^ r)) by FINSEQ_1:def 3;
then reconsider Fp = ((L1 + L2) (#) (F ^ r)) * P as FinSequence of the carrier of V by FINSEQ_2:47;
A59: len (L1 (#) (G ^ p)) = len (G ^ p) by Def22;
deffunc H₂( Nat) -> set = (H ^ q) <- ((G ^ p) . $1);
consider R being FinSequence such that
A60: len R = len (H ^ q) and
A61: for k being Nat st k in dom R holds
R . k = H₂(k) from FINSEQ_1:sch 2();
A62: dom R = Seg (len (H ^ q)) by A60, FINSEQ_1:def 3;
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 A19, A21, XBOOLE_1:39;
then A63: rng (H ^ q) = ((Carrier (L1 + L2)) \/ (Carrier L1)) \/ (Carrier L2) by XBOOLE_1:7, XBOOLE_1:12;
then A64: len (G ^ p) = len (H ^ q) by A23, A40, A25, FINSEQ_1:48;
A65: dom R = Seg (len (H ^ q)) by A60, FINSEQ_1:def 3;

A66: now :: thesis:

let x be object ; :: thesis:
assume A67: x in dom (G ^ p) ; :: thesis:
then reconsider n = x as Element of NAT ;
(G ^ p) . n in rng (H ^ q) by A25, A63, A67, FUNCT_1:def 3;
then A68: H ^ q just_once_values (G ^ p) . n by A23, FINSEQ_4:8;
n in Seg (len (H ^ q)) by A64, A67, FINSEQ_1:def 3;
then (H ^ q) . (R . n) = (H ^ q) . ((H ^ q) <- ((G ^ p) . n)) by A61, A65
.= (G ^ p) . n by A68, FINSEQ_4:def 3 ;
hence (G ^ p) . x = (H ^ q) . (R . x) ; :: thesis:

end;

A69: rng R c= Seg (len (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
A70: y in dom R and
A71: R . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A70;
y in Seg (len (H ^ q)) by A60, A70, FINSEQ_1:def 3;
then y in dom (G ^ p) by A64, FINSEQ_1:def 3;
then (G ^ p) . y in rng (H ^ q) by A25, A63, FUNCT_1:def 3;
then A72: H ^ q just_once_values (G ^ p) . y by A23, FINSEQ_4:8;
R . y = (H ^ q) <- ((G ^ p) . y) by A61, A70;
then R . y in dom (H ^ q) by A72, FINSEQ_4:def 3;
hence x in Seg (len (H ^ q)) by A71, FINSEQ_1: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:
then x in Seg (len R) by A64, A60, FINSEQ_1:def 3;
hence x in dom R by FINSEQ_1:def 3; :: thesis:
then R . x in rng R by FUNCT_1:def 3;
then R . x in Seg (len (H ^ q)) by A69;
hence R . x in dom (H ^ q) by FINSEQ_1:def 3; :: thesis:

end;

assume that
A73: x in dom R and
R . x in dom (H ^ q) ; :: thesis:
x in Seg (len R) by A73, FINSEQ_1:def 3;
hence x in dom (G ^ p) by A64, A60, FINSEQ_1:def 3; :: thesis:

end;

then A74: G ^ p = (H ^ q) * R by A66, FUNCT_1:10;
Seg (len (H ^ q)) c= rng R

proof

set f = ((H ^ q) ") * (G ^ p);

now :: thesis:

let x be object ; :: thesis:
assume A75: x in Seg (len (H ^ q)) ; :: thesis:
dom ((H ^ q) ") = rng (G ^ p) by A23, A25, A63, FUNCT_1:33;
then A76: rng (((H ^ q) ") * (G ^ p)) = rng ((H ^ q) ") by RELAT_1:28
.= dom (H ^ q) by A23, FUNCT_1:33 ;
A77: rng R c= dom (H ^ q) by A69, FINSEQ_1:def 3;
((H ^ q) ") * (G ^ p) = (((H ^ q) ") * (H ^ q)) * R by A74, RELAT_1:36
.= (id (dom (H ^ q))) * R by A23, FUNCT_1:39
.= R by A77, RELAT_1:53 ;
hence x in rng R by A75, A76, FINSEQ_1:def 3; :: thesis:

end;

hence Seg (len (H ^ q)) c= rng R ; :: thesis:

end;

then A78: Seg (len (H ^ q)) = rng R by A69;
then A79: R is one-to-one by A62, FINSEQ_4:60;
reconsider R = R as Function of (Seg (len (H ^ q))),(Seg (len (H ^ q))) by FUNCT_2:1, A78, A65;
reconsider R = R as Permutation of (Seg (len (H ^ q))) by A78, A79, FUNCT_2:57;
A80: len (L2 (#) (H ^ q)) = len (H ^ q) by Def22;
then A81: Seg (len (H ^ q)) = dom (L2 (#) (H ^ q)) by FINSEQ_1:def 3;
then reconsider Hp = (L2 (#) (H ^ q)) * R as FinSequence of the carrier of V by FINSEQ_2:47;
A82: len Hp = len (G ^ p) by A64, A80, A81, FINSEQ_2:44;
deffunc H₃( Nat) -> Element of the carrier of V = ((L1 (#) (G ^ p)) /. $1) + (Hp /. $1);
consider I being FinSequence such that
A83: len I = len (G ^ p) and
A84: for k being Nat st k in dom I holds
I . k = H₃(k) from FINSEQ_1:sch 2();
dom I = Seg (len (G ^ p)) by A83, FINSEQ_1:def 3;
then A85: for k being Nat st k in Seg (len (G ^ p)) holds
I . k = H₃(k) by A84;
rng I c= the carrier of V

proof

let x be object ; :: according to TARSKI:def 3 :: thesis:
assume x in rng I ; :: thesis:
then consider y being object such that
A86: y in dom I and
A87: I . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A86;
I . y = ((L1 (#) (G ^ p)) /. y) + (Hp /. y) by A84, A86;
hence x in the carrier of V by A87; :: thesis:

end;

then reconsider I = I as FinSequence of the carrier of V by FINSEQ_1:def 4;
A88: len Fp = len I by A41, A57, A58, A83, FINSEQ_2:44;

A89: now :: thesis:

let x be object ; :: thesis:
assume A90: x in dom I ; :: thesis:
then reconsider k = x as Element of NAT ;
A91: x in dom Hp by A83, A82, A90, FINSEQ_3:29;
k in dom R by A64, A62, A83, A90, FINSEQ_1:def 3;
then A92: R . k in dom R by A78, A62, FUNCT_1:def 3;
then reconsider j = R . k as Element of NAT by FINSEQ_3:23;
set v = (G ^ p) /. k;
A93: R . k in dom (H ^ q) by A60, A92, FINSEQ_3:29;
A94: x in dom (G ^ p) by A83, A90, FINSEQ_3:29;
then (H ^ q) . j = (G ^ p) . k by A66
.= (G ^ p) /. k by A94, PARTFUN1:def 6 ;
then A95: (L2 (#) (H ^ q)) . j = (L2 . ((G ^ p) /. k)) * ((G ^ p) /. k) by A93, Th13;
k in dom P by A41, A33, A83, A90, FINSEQ_1:def 3;
then A96: P . k in dom P by A55, A33, FUNCT_1:def 3;
then reconsider l = P . k as Element of NAT by FINSEQ_3:23;
A97: P . k in dom (F ^ r) by A31, A96, FINSEQ_3:29;
x in dom Fp by A88, A90, FINSEQ_3:29;
then A98: Fp . k = ((L1 + L2) (#) (F ^ r)) . (P . k) by FUNCT_1:12;
k in dom Hp by A83, A82, A90, FINSEQ_3:29;
then A99: Hp /. k = ((L2 (#) (H ^ q)) * R) . k by PARTFUN1:def 6
.= (L2 (#) (H ^ q)) . (R . k) by A91, FUNCT_1:12 ;
A100: x in dom (L1 (#) (G ^ p)) by A83, A59, A90, FINSEQ_3:29;
(F ^ r) . l = (G ^ p) . k by A43, A94
.= (G ^ p) /. k by A94, PARTFUN1:def 6 ;
then A101: ((L1 + L2) (#) (F ^ r)) . l = ((L1 + L2) . ((G ^ p) /. k)) * ((G ^ p) /. k) by A97, Th13
.= ((L1 . ((G ^ p) /. k)) + (L2 . ((G ^ p) /. k))) * ((G ^ p) /. k) by Def25
.= ((L1 . ((G ^ p) /. k)) * ((G ^ p) /. k)) + ((L2 . ((G ^ p) /. k)) * ((G ^ p) /. k)) by ZMODUL01:def 3 ;
k in dom (L1 (#) (G ^ p)) by A83, A59, A90, FINSEQ_3:29;
then (L1 (#) (G ^ p)) /. k = (L1 (#) (G ^ p)) . k by PARTFUN1:def 6
.= (L1 . ((G ^ p) /. k)) * ((G ^ p) /. k) by A100, Def22 ;
hence I . x = Fp . x by A84, A90, A99, A95, A98, A101; :: thesis:

end;

dom (L2 (#) (H ^ q)) = Seg (len (L2 (#) (H ^ q))) by FINSEQ_1:def 3;
then A102: Sum Hp = Sum (L2 (#) (H ^ q)) by A80, RLVECT_2:7;
dom ((L1 + L2) (#) (F ^ r)) = Seg (len ((L1 + L2) (#) (F ^ r))) by FINSEQ_1:def 3;
then A103: Sum Fp = Sum ((L1 + L2) (#) (F ^ r)) by A57, RLVECT_2:7;
( dom I = Seg (len I) & dom Fp = Seg (len I) ) by A88, FINSEQ_1:def 3;
then A104: I = Fp by A89;
Seg (len (G ^ p)) = dom (L1 (#) (G ^ p)) by A59, FINSEQ_1:def 3;
hence Sum (L1 + L2) = (Sum L1) + (Sum L2) by A3, A14, A20, A38, A30, A17, A103, A102, A83, A85, A82, A59, A104, RLVECT_2:2; :: thesis: