let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF

for L1, L2 being Linear_Combination of V holds Sum (L1 + L2) = (Sum L1) + (Sum L2)

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for L1, L2 being Linear_Combination of V holds Sum (L1 + L2) = (Sum L1) + (Sum L2)

let L1, L2 be Linear_Combination of V; :: thesis: Sum (L1 + L2) = (Sum L1) + (Sum L2)

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 the carrier of V 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 the carrier of V 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 the carrier of V by A7, FINSEQ_1:def 4;

consider F being FinSequence of V such that

A9: F is one-to-one and

A10: rng F = Carrier (L1 + L2) and

A11: Sum ((L1 + L2) (#) F) = Sum (L1 + L2) by Def6;

set FF = F ^ r;

consider G being FinSequence of V such that

A12: G is one-to-one and

A13: rng G = Carrier L1 and

A14: Sum (L1 (#) G) = Sum L1 by Def6;

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

rng G misses rng p

then A19: len (G ^ p) = len (F ^ r) by A17, A16, A15, FINSEQ_1:48;

deffunc H_{1}( 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 = H_{1}(k)
from FINSEQ_1:sch 2();

A22: dom P = Seg (len (F ^ r)) by A20, FINSEQ_1:def 3;

dom (F ^ r) c= rng P

A35: len r = len ((L1 + L2) (#) r) by Def5;

.= 0. V by VECTSP_1:14 ;

A39: len p = len (L1 (#) p) by Def5;

.= 0. V by VECTSP_1:14 ;

A43: len q = len (L2 (#) q) by Def5;

.= 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

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 H_{2}( 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 = H_{2}(k)
from FINSEQ_1:sch 2();

A57: dom R = Seg (len (H ^ q)) by A55, FINSEQ_1:def 3;

dom (H ^ q) c= rng R

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 H_{3}( 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 = H_{3}(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 = H_{3}(k)
by A81;

rng I c= the carrier of V

A86: len Fp = len I by A19, A78, A80, FINSEQ_2:44;

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: verum

for L1, L2 being Linear_Combination of V holds Sum (L1 + L2) = (Sum L1) + (Sum L2)

let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for L1, L2 being Linear_Combination of V holds Sum (L1 + L2) = (Sum L1) + (Sum L2)

let L1, L2 be Linear_Combination of V; :: thesis: Sum (L1 + L2) = (Sum L1) + (Sum L2)

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 the carrier of V 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 the carrier of V 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 the carrier of V by A7, FINSEQ_1:def 4;

consider F being FinSequence of V such that

A9: F is one-to-one and

A10: rng F = Carrier (L1 + L2) and

A11: Sum ((L1 + L2) (#) F) = Sum (L1 + L2) by Def6;

set FF = F ^ r;

consider G being FinSequence of V such that

A12: G is one-to-one and

A13: rng G = Carrier L1 and

A14: Sum (L1 (#) G) = Sum L1 by Def6;

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

then A17:
F ^ r is one-to-one
by A9, A5, FINSEQ_3:91;
set x = the Element of (rng F) /\ (rng r);

assume not rng F misses rng r ; :: thesis: contradiction

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: verum

end;assume not rng F misses rng r ; :: thesis: contradiction

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: verum

rng G misses rng p

proof

then A18:
G ^ p is one-to-one
by A12, A2, FINSEQ_3:91;
set x = the Element of (rng G) /\ (rng p);

assume not rng G misses rng p ; :: thesis: contradiction

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: verum

end;assume not rng G misses rng p ; :: thesis: contradiction

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: verum

then A19: len (G ^ p) = len (F ^ r) by A17, A16, A15, FINSEQ_1:48;

deffunc H

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 = H

A22: dom P = Seg (len (F ^ r)) by A20, FINSEQ_1:def 3;

A23: now :: thesis: for x being object st x in dom (G ^ p) holds

(G ^ p) . x = (F ^ r) . (P . x)

A26:
rng P c= dom (F ^ r)
(G ^ p) . x = (F ^ r) . (P . x)

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

assume A24: x in dom (G ^ p) ; :: thesis: (G ^ p) . x = (F ^ r) . (P . x)

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: verum

end;assume A24: x in dom (G ^ p) ; :: thesis: (G ^ p) . x = (F ^ r) . (P . x)

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: verum

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng P or x in dom (F ^ r) )

assume x in rng P ; :: thesis: x in dom (F ^ r)

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: verum

end;assume x in rng P ; :: thesis: x in dom (F ^ r)

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: verum

now :: thesis: for x being object holds

( ( x in dom (G ^ p) implies ( x in dom P & P . x in dom (F ^ r) ) ) & ( x in dom P & P . x in dom (F ^ r) implies x in dom (G ^ p) ) )

then A31:
G ^ p = (F ^ r) * P
by A23, FUNCT_1:10;( ( x in dom (G ^ p) implies ( x in dom P & P . x in dom (F ^ r) ) ) & ( x in dom P & P . x in dom (F ^ r) implies x in dom (G ^ p) ) )

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

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

A30: x in dom P and

P . x in dom (F ^ r) ; :: thesis: x in dom (G ^ p)

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: verum

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

proof

assume that
assume
x in dom (G ^ p)
; :: thesis: ( x in dom P & P . x in dom (F ^ r) )

then x in Seg (len P) by A19, A20, FINSEQ_1:def 3;

hence x in dom P by FINSEQ_1:def 3; :: thesis: P . x in dom (F ^ r)

then P . x in rng P by FUNCT_1:def 3;

hence P . x in dom (F ^ r) by A26; :: thesis: verum

end;then x in Seg (len P) by A19, A20, FINSEQ_1:def 3;

hence x in dom P by FINSEQ_1:def 3; :: thesis: P . x in dom (F ^ r)

then P . x in rng P by FUNCT_1:def 3;

hence P . x in dom (F ^ r) by A26; :: thesis: verum

A30: x in dom P and

P . x in dom (F ^ r) ; :: thesis: x in dom (G ^ p)

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: verum

dom (F ^ r) c= rng P

proof

then A34:
dom (F ^ r) = rng P
by A26;
set f = ((F ^ r) ") * (G ^ p);

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in dom (F ^ r) or x in rng P )

assume A32: x in dom (F ^ r) ; :: thesis: x in rng P

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: verum

end;let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in dom (F ^ r) or x in rng P )

assume A32: x in dom (F ^ r) ; :: thesis: x in rng P

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: verum

A35: len r = len ((L1 + L2) (#) r) by Def5;

now :: thesis: for k being Nat st k in dom r holds

((L1 + L2) (#) r) . k = (0. GF) * (r /. k)

then A38: Sum ((L1 + L2) (#) r) =
(0. GF) * (Sum r)
by A35, RLVECT_2:67
((L1 + L2) (#) r) . k = (0. GF) * (r /. k)

let k be Nat; :: thesis: ( k in dom r implies ((L1 + L2) (#) r) . k = (0. GF) * (r /. k) )

assume A36: k in dom r ; :: thesis: ((L1 + L2) (#) r) . k = (0. GF) * (r /. k)

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: verum

end;assume A36: k in dom r ; :: thesis: ((L1 + L2) (#) r) . k = (0. GF) * (r /. k)

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: verum

.= 0. V by VECTSP_1:14 ;

A39: len p = len (L1 (#) p) by Def5;

now :: thesis: for k being Nat st k in dom p holds

(L1 (#) p) . k = (0. GF) * (p /. k)

then A42: Sum (L1 (#) p) =
(0. GF) * (Sum p)
by A39, RLVECT_2:67
(L1 (#) p) . k = (0. GF) * (p /. k)

let k be Nat; :: thesis: ( k in dom p implies (L1 (#) p) . k = (0. GF) * (p /. k) )

assume A40: k in dom p ; :: thesis: (L1 (#) p) . k = (0. GF) * (p /. k)

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: verum

end;assume A40: k in dom p ; :: thesis: (L1 (#) p) . k = (0. GF) * (p /. k)

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: verum

.= 0. V by VECTSP_1:14 ;

A43: len q = len (L2 (#) q) by Def5;

now :: thesis: for k being Nat st k in dom q holds

(L2 (#) q) . k = (0. GF) * (q /. k)

then A46: Sum (L2 (#) q) =
(0. GF) * (Sum q)
by A43, RLVECT_2:67
(L2 (#) q) . k = (0. GF) * (q /. k)

let k be Nat; :: thesis: ( k in dom q implies (L2 (#) q) . k = (0. GF) * (q /. k) )

assume A44: k in dom q ; :: thesis: (L2 (#) q) . k = (0. GF) * (q /. k)

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: verum

end;assume A44: k in dom q ; :: thesis: (L2 (#) q) . k = (0. GF) * (q /. k)

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: verum

.= 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

then A52:
H ^ q is one-to-one
by A49, A8, FINSEQ_3:91;
set x = the Element of (rng H) /\ (rng q);

assume not rng H misses rng q ; :: thesis: contradiction

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: verum

end;assume not rng H misses rng q ; :: thesis: contradiction

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: verum

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 H

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 = H

A57: dom R = Seg (len (H ^ q)) by A55, FINSEQ_1:def 3;

A58: now :: thesis: for x being object st x in dom (G ^ p) holds

(G ^ p) . x = (H ^ q) . (R . x)

A61:
rng R c= dom (H ^ q)
(G ^ p) . x = (H ^ q) . (R . x)

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

assume A59: x in dom (G ^ p) ; :: thesis: (G ^ p) . x = (H ^ q) . (R . x)

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: verum

end;assume A59: x in dom (G ^ p) ; :: thesis: (G ^ p) . x = (H ^ q) . (R . x)

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: verum

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng R or x in dom (H ^ q) )

assume x in rng R ; :: thesis: x in dom (H ^ q)

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: verum

end;assume x in rng R ; :: thesis: x in dom (H ^ q)

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: verum

now :: thesis: for x being object holds

( ( x in dom (G ^ p) implies ( x in dom R & R . x in dom (H ^ q) ) ) & ( x in dom R & R . x in dom (H ^ q) implies x in dom (G ^ p) ) )

then A66:
G ^ p = (H ^ q) * R
by A58, FUNCT_1:10;( ( x in dom (G ^ p) implies ( x in dom R & R . x in dom (H ^ q) ) ) & ( x in dom R & R . x in dom (H ^ q) implies x in dom (G ^ p) ) )

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

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

A65: x in dom R and

R . x in dom (H ^ q) ; :: thesis: x in dom (G ^ p)

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: verum

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

proof

assume that
assume
x in dom (G ^ p)
; :: thesis: ( x in dom R & R . x in dom (H ^ q) )

then x in Seg (len R) by A54, A55, FINSEQ_1:def 3;

hence x in dom R by FINSEQ_1:def 3; :: thesis: R . x in dom (H ^ q)

then R . x in rng R by FUNCT_1:def 3;

hence R . x in dom (H ^ q) by A61; :: thesis: verum

end;then x in Seg (len R) by A54, A55, FINSEQ_1:def 3;

hence x in dom R by FINSEQ_1:def 3; :: thesis: R . x in dom (H ^ q)

then R . x in rng R by FUNCT_1:def 3;

hence R . x in dom (H ^ q) by A61; :: thesis: verum

A65: x in dom R and

R . x in dom (H ^ q) ; :: thesis: x in dom (G ^ p)

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: verum

dom (H ^ q) c= rng R

proof

then A69:
dom (H ^ q) = rng R
by A61;
set f = ((H ^ q) ") * (G ^ p);

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in dom (H ^ q) or x in rng R )

assume A67: x in dom (H ^ q) ; :: thesis: x in rng R

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: verum

end;let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in dom (H ^ q) or x in rng R )

assume A67: x in dom (H ^ q) ; :: thesis: x in rng R

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: verum

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 H

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 = H

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 = H

rng I c= the carrier of V

proof

then reconsider I = I as FinSequence of the carrier of V by FINSEQ_1:def 4;
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng I or x in the carrier of V )

assume x in rng I ; :: thesis: x in the carrier of V

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: verum

end;assume x in rng I ; :: thesis: x in the carrier of V

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: verum

A86: len Fp = len I by A19, A78, A80, FINSEQ_2:44;

A87: now :: thesis: for x being object st x in Seg (len I) holds

I . x = Fp . x

( dom I = Seg (len I) & dom Fp = Seg (len I) )
by A86, FINSEQ_1:def 3;I . x = Fp . x

let x be object ; :: thesis: ( x in Seg (len I) implies I . x = Fp . x )

assume A88: x in Seg (len I) ; :: thesis: I . x = Fp . x

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: verum

end;assume A88: x in Seg (len I) ; :: thesis: I . x = Fp . x

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: verum

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: verum