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 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 (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, 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 by FINSEQ_3:23;
(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, FINSEQ_3:23;
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 ((L1 + L2) (#) r) by Def5;

now :: thesis:

let k be Nat; :: thesis:
assume A36: 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, A36, FUNCT_1:def 3;
then A37: not r /. k in Carrier (L1 + L2) by XBOOLE_0:def 5;
k in dom ((L1 + L2) (#) r) by A35, A36, FINSEQ_3:29;
then ((L1 + L2) (#) r) . k = ((L1 + L2) . (r /. k)) * (r /. k) by Def5;
hence ((L1 + L2) (#) r) . k = (0. GF) * (r /. k) by A37; :: thesis:

end;

then A38: Sum ((L1 + L2) (#) r) = (0. GF) * (Sum r) by A35, RLVECT_2:67
.= 0. V by VECTSP_1:14 ;
A39: len p = len (L1 (#) p) by Def5;

now :: thesis:

let k be Nat; :: thesis:
assume A40: 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 A1, A40, FUNCT_1:def 3;
then A41: not p /. k in Carrier L1 by XBOOLE_0:def 5;
k in dom (L1 (#) p) by A39, A40, FINSEQ_3:29;
then (L1 (#) p) . k = (L1 . (p /. k)) * (p /. k) by Def5;
hence (L1 (#) p) . k = (0. GF) * (p /. k) by A41; :: thesis:

end;

then A42: Sum (L1 (#) p) = (0. GF) * (Sum p) by A39, RLVECT_2:67
.= 0. V by VECTSP_1:14 ;
A43: len q = len (L2 (#) q) by Def5;

now :: thesis:

let k be Nat; :: thesis:
assume A44: 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 A7, A44, FUNCT_1:def 3;
then A45: not q /. k in Carrier L2 by XBOOLE_0:def 5;
k in dom (L2 (#) q) by A43, A44, FINSEQ_3:29;
then (L2 (#) q) . k = (L2 . (q /. k)) * (q /. k) by Def5;
hence (L2 (#) q) . k = (0. GF) * (q /. k) by A45; :: thesis:

end;

then A46: Sum (L2 (#) q) = (0. GF) * (Sum q) by A43, RLVECT_2:67
.= 0. V by VECTSP_1:14 ;
set g = L1 (#) (G ^ p);
A47: len (L1 (#) (G ^ p)) = len (G ^ p) by Def5;
then A48: Seg (len (G ^ p)) = dom (L1 (#) (G ^ p)) by FINSEQ_1:def 3;
set f = (L1 + L2) (#) (F ^ r);
consider H being FinSequence of V such that
A49: H is one-to-one and
A50: rng H = Carrier L2 and
A51: Sum (L2 (#) H) = Sum L2 by Def6;
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 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;
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 = Seg (len (H ^ q)) by A55, FINSEQ_1:def 3;

A58: now :: thesis:

let x be object ; :: thesis:
assume A59: x in dom (G ^ p) ; :: thesis:
then reconsider n = x as Element of NAT by FINSEQ_3:23;
(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;
n in Seg (len (H ^ q)) by A54, A59, FINSEQ_1:def 3;
then (H ^ q) . (R . n) = (H ^ q) . ((H ^ q) <- ((G ^ p) . n)) by A56, A57
.= (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, FINSEQ_3:23;
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:
then x in Seg (len R) by A54, A55, 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;
hence R . x in dom (H ^ q) by A61; :: thesis:

end;

assume that
A65: x in dom R and
R . x in dom (H ^ q) ; :: thesis:
x in Seg (len R) by A65, FINSEQ_1:def 3;
hence x in dom (G ^ p) by A54, A55, FINSEQ_1:def 3; :: 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 = L2 (#) (H ^ q);
A70: Sum (L2 (#) (H ^ q)) = Sum ((L2 (#) H) ^ (L2 (#) q)) by Th13
.= (Sum (L2 (#) H)) + (0. V) by A46, RLVECT_1:41
.= Sum (L2 (#) H) by RLVECT_1:4 ;
A71: Sum (L1 (#) (G ^ p)) = Sum ((L1 (#) G) ^ (L1 (#) p)) by Th13
.= (Sum (L1 (#) G)) + (0. V) by A42, RLVECT_1:41
.= Sum (L1 (#) G) by RLVECT_1:4 ;
A72: dom P = dom (F ^ r) by A20, FINSEQ_3:29;
then A73: P is one-to-one by A34, FINSEQ_4:60;
A74: dom R = dom (H ^ q) by A55, FINSEQ_3:29;
then A75: R is one-to-one by A69, FINSEQ_4:60;
reconsider R = R as Function of (dom (H ^ q)),(dom (H ^ q)) by A61, A74, FUNCT_2:2;
reconsider R = R as Permutation of (dom (H ^ q)) by A69, A75, FUNCT_2:57;
A76: len (L2 (#) (H ^ q)) = len (H ^ q) by Def5;
then dom (L2 (#) (H ^ q)) = dom (H ^ q) by FINSEQ_3:29;
then reconsider R = R as Permutation of (dom (L2 (#) (H ^ q))) ;
reconsider Hp = (L2 (#) (H ^ q)) * R as FinSequence of the carrier of V 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 ((L1 + L2) (#) (F ^ r)) = len (F ^ r) by Def5;
then dom ((L1 + L2) (#) (F ^ r)) = dom (F ^ r) by FINSEQ_3:29;
then reconsider P = P as Permutation of (dom ((L1 + L2) (#) (F ^ r))) ;
reconsider Fp = ((L1 + L2) (#) (F ^ r)) * P as FinSequence of the carrier of V by FINSEQ_2:47;
A79: Sum ((L1 + L2) (#) (F ^ r)) = Sum (((L1 + L2) (#) F) ^ ((L1 + L2) (#) r)) by Th13
.= (Sum ((L1 + L2) (#) F)) + (0. V) by A38, RLVECT_1:41
.= Sum ((L1 + L2) (#) F) by RLVECT_1:4 ;
deffunc H₃( Nat) -> Element of the carrier of V = ((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();
A82: dom I = Seg (len (G ^ p)) by A80, FINSEQ_1:def 3;
then A83: for k being Nat st k in Seg (len (G ^ p)) holds
I . k = H₃(k) by A81;
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
A84: y in dom I and
A85: I . y = x by FUNCT_1:def 3;
reconsider y = y as Element of NAT by A84, FINSEQ_3:23;
I . y = ((L1 (#) (G ^ p)) /. y) + (Hp /. y) by A81, A84;
hence x in the carrier of V by A85; :: thesis:

end;

then reconsider I = I as FinSequence of the carrier of V by FINSEQ_1:def 4;
A86: len Fp = len I by A19, A78, A80, FINSEQ_2:44;

A87: now :: thesis:

let x be object ; :: thesis:
assume A88: x in Seg (len I) ; :: thesis:
then reconsider k = x as Element of NAT ;
A89: x in dom Hp by A80, A77, A88, FINSEQ_1:def 3;
then A90: Hp /. k = ((L2 (#) (H ^ q)) * R) . k by PARTFUN1:def 6
.= (L2 (#) (H ^ q)) . (R . k) by A89, FUNCT_1:12 ;
set v = (G ^ p) /. k;
x in dom Fp by A86, A88, FINSEQ_1:def 3;
then A91: Fp . k = ((L1 + L2) (#) (F ^ r)) . (P . k) by FUNCT_1:12;
A92: x in dom (H ^ q) by A54, A80, A88, FINSEQ_1:def 3;
then R . k in dom R by A69, A74, FUNCT_1:def 3;
then reconsider j = R . k as Element of NAT by FINSEQ_3:23;
A93: x in dom (G ^ p) by A80, A88, FINSEQ_1:def 3;
then A94: (H ^ q) . j = (G ^ p) . k by A58
.= (G ^ p) /. k by A93, PARTFUN1:def 6 ;
A95: dom (F ^ r) = dom (G ^ p) by A19, FINSEQ_3:29;
then P . k in dom P by A34, A72, A93, FUNCT_1:def 3;
then reconsider l = P . k as Element of NAT by FINSEQ_3:23;
A96: (F ^ r) . l = (G ^ p) . k by A23, A93
.= (G ^ p) /. k by A93, PARTFUN1:def 6 ;
R . k in dom (H ^ q) by A69, A74, A92, FUNCT_1:def 3;
then A97: (L2 (#) (H ^ q)) . j = (L2 . ((G ^ p) /. k)) * ((G ^ p) /. k) by A94, Th8;
P . k in dom (F ^ r) by A34, A72, A93, A95, FUNCT_1:def 3;
then A98: ((L1 + L2) (#) (F ^ r)) . l = ((L1 + L2) . ((G ^ p) /. k)) * ((G ^ p) /. k) by A96, Th8
.= ((L1 . ((G ^ p) /. k)) + (L2 . ((G ^ p) /. k))) * ((G ^ p) /. k) by Th22
.= ((L1 . ((G ^ p) /. k)) * ((G ^ p) /. k)) + ((L2 . ((G ^ p) /. k)) * ((G ^ p) /. k)) by VECTSP_1:def 15 ;
A99: x in dom (L1 (#) (G ^ p)) by A80, A47, A88, FINSEQ_1:def 3;
then (L1 (#) (G ^ p)) /. k = (L1 (#) (G ^ p)) . k by PARTFUN1:def 6
.= (L1 . ((G ^ p) /. k)) * ((G ^ p) /. k) by A99, Def5 ;
hence I . x = Fp . x by A80, A81, A82, 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 ((L1 + L2) (#) (F ^ r)) & Sum Hp = Sum (L2 (#) (H ^ q)) ) by RLVECT_2:7;
hence Sum (L1 + L2) = (Sum L1) + (Sum L2) by A11, A14, A51, A71, A70, A79, A80, A83, A77, A47, A100, A48, RLVECT_2:2; :: thesis: