:: Linear Combinations in Real Linear Space
:: by Wojciech A. Trybulec
::
:: Copyright (c) 1990-2021 Association of Mizar Users

definition
let S be 1-sorted ;
let x be set ;
assume A1: x in S ;
func vector (S,x) -> Element of S equals :Def1: :: RLVECT_2:def 1
x;
coherence
x is Element of S
by A1;
end;

:: deftheorem Def1 defines vector RLVECT_2:def 1 :
for S being 1-sorted
for x being set st x in S holds
vector (S,x) = x;

theorem :: RLVECT_2:1
for S being non empty 1-sorted
for v being Element of S holds vector (S,v) = v by ;

theorem Th2: :: RLVECT_2:2
for F, G, H being FinSequence of the carrier of V st len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) + (G /. k) ) holds
Sum H = (Sum F) + (Sum G)
proof end;

theorem :: RLVECT_2:3
for V being RealLinearSpace
for a being Real
for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)
proof end;

theorem Th4: :: RLVECT_2:4
for F, G being FinSequence of the carrier of V st len F = len G & ( for k being Nat st k in dom F holds
G . k = - (F /. k) ) holds
Sum G = - (Sum F)
proof end;

theorem :: RLVECT_2:5
for F, G, H being FinSequence of the carrier of V st len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)
proof end;

theorem Th6: :: RLVECT_2:6
for F, G being FinSequence of the carrier of V
for f being Permutation of (dom F) st len F = len G & ( for i being Nat st i in dom G holds
G . i = F . (f . i) ) holds
Sum F = Sum G
proof end;

theorem :: RLVECT_2:7
for F, G being FinSequence of the carrier of V
for f being Permutation of (dom F) st G = F * f holds
Sum F = Sum G
proof end;

definition
let V be non empty addLoopStr ;
let T be finite Subset of V;
assume A1: ( V is Abelian & V is add-associative & V is right_zeroed ) ;
func Sum T -> Element of V means :Def2: :: RLVECT_2:def 2
ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & it = Sum F );
existence
ex b1 being Element of V ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b1 = Sum F )
proof end;
uniqueness
for b1, b2 being Element of V st ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b1 = Sum F ) & ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b2 = Sum F ) holds
b1 = b2
by ;
end;

:: deftheorem Def2 defines Sum RLVECT_2:def 2 :
for V being non empty addLoopStr
for T being finite Subset of V st V is Abelian & V is add-associative & V is right_zeroed holds
for b3 being Element of V holds
( b3 = Sum T iff ex F being FinSequence of the carrier of V st
( rng F = T & F is one-to-one & b3 = Sum F ) );

theorem Th8: :: RLVECT_2:8
for V being non empty Abelian add-associative right_zeroed addLoopStr holds Sum ({} V) = 0. V
proof end;

theorem :: RLVECT_2:9
for v being Element of V holds Sum {v} = v
proof end;

theorem :: RLVECT_2:10
for v1, v2 being Element of V st v1 <> v2 holds
Sum {v1,v2} = v1 + v2
proof end;

theorem :: RLVECT_2:11
for v1, v2, v3 being Element of V st v1 <> v2 & v2 <> v3 & v1 <> v3 holds
Sum {v1,v2,v3} = (v1 + v2) + v3
proof end;

theorem Th12: :: RLVECT_2:12
for S, T being finite Subset of V st T misses S holds
Sum (T \/ S) = (Sum T) + (Sum S)
proof end;

theorem Th13: :: RLVECT_2:13
for S, T being finite Subset of V holds Sum (T \/ S) = ((Sum T) + (Sum S)) - (Sum (T /\ S))
proof end;

theorem :: RLVECT_2:14
for S, T being finite Subset of V holds Sum (T /\ S) = ((Sum T) + (Sum S)) - (Sum (T \/ S))
proof end;

theorem Th15: :: RLVECT_2:15
for S, T being finite Subset of V holds Sum (T \ S) = (Sum (T \/ S)) - (Sum S)
proof end;

theorem Th16: :: RLVECT_2:16
for S, T being finite Subset of V holds Sum (T \ S) = (Sum T) - (Sum (T /\ S))
proof end;

theorem :: RLVECT_2:17
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \/ S)) - (Sum (T /\ S))
proof end;

theorem :: RLVECT_2:18
for S, T being finite Subset of V holds Sum (T \+\ S) = (Sum (T \ S)) + (Sum (S \ T)) by ;

reconsider zz = 0 as Element of REAL by XREAL_0:def 1;

definition
let V be non empty ZeroStr ;
mode Linear_Combination of V -> Element of Funcs ( the carrier of V,REAL) means :Def3: :: RLVECT_2:def 3
ex T being finite Subset of V st
for v being Element of V st not v in T holds
it . v = 0 ;
existence
ex b1 being Element of Funcs ( the carrier of V,REAL) ex T being finite Subset of V st
for v being Element of V st not v in T holds
b1 . v = 0
proof end;
end;

:: deftheorem Def3 defines Linear_Combination RLVECT_2:def 3 :
for V being non empty ZeroStr
for b2 being Element of Funcs ( the carrier of V,REAL) holds
( b2 is Linear_Combination of V iff ex T being finite Subset of V st
for v being Element of V st not v in T holds
b2 . v = 0 );

notation
let V be non empty addLoopStr ;
let L be Element of Funcs ( the carrier of V,REAL);
synonym Carrier L for support V;
end;

Lm1: now :: thesis: for V being non empty addLoopStr
for L being Element of Funcs ( the carrier of V,REAL) holds Carrier c= the carrier of V
let V be non empty addLoopStr ; :: thesis: for L being Element of Funcs ( the carrier of V,REAL) holds Carrier c= the carrier of V
let L be Element of Funcs ( the carrier of V,REAL); :: thesis: Carrier c= the carrier of V
A1: support L c= dom L by PRE_POLY:37;
thus Carrier c= the carrier of V :: thesis: verum
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in Carrier or x in the carrier of V )
assume x in support L ; :: thesis: x in the carrier of V
then x in dom L by A1;
hence x in the carrier of V ; :: thesis: verum
end;
end;

definition
let V be non empty addLoopStr ;
let L be Element of Funcs ( the carrier of V,REAL);
:: original: Carrier
redefine func Carrier L -> Subset of V equals :: RLVECT_2:def 4
{ v where v is Element of V : L . v <> 0 } ;
coherence
Carrier is Subset of V
by Lm1;
compatibility
for b1 being Subset of V holds
( b1 = Carrier iff b1 = { v where v is Element of V : L . v <> 0 } )
proof end;
end;

:: deftheorem defines Carrier RLVECT_2:def 4 :
for V being non empty addLoopStr
for L being Element of Funcs ( the carrier of V,REAL) holds Carrier L = { v where v is Element of V : L . v <> 0 } ;

registration
let V be non empty addLoopStr ;
let L be Linear_Combination of V;
coherence
proof end;
end;

theorem :: RLVECT_2:19
for V being non empty addLoopStr
for L being Linear_Combination of V
for v being Element of V holds
( L . v = 0 iff not v in Carrier L )
proof end;

definition
let V be non empty addLoopStr ;
func ZeroLC V -> Linear_Combination of V means :Def5: :: RLVECT_2:def 5
Carrier it = {} ;
existence
ex b1 being Linear_Combination of V st Carrier b1 = {}
proof end;
uniqueness
for b1, b2 being Linear_Combination of V st Carrier b1 = {} & Carrier b2 = {} holds
b1 = b2
proof end;
end;

:: deftheorem Def5 defines ZeroLC RLVECT_2:def 5 :
for V being non empty addLoopStr
for b2 being Linear_Combination of V holds
( b2 = ZeroLC V iff Carrier b2 = {} );

theorem Th20: :: RLVECT_2:20
for V being non empty addLoopStr
for v being Element of V holds () . v = 0
proof end;

definition
let V be non empty addLoopStr ;
let A be Subset of V;
mode Linear_Combination of A -> Linear_Combination of V means :Def6: :: RLVECT_2:def 6
Carrier it c= A;
existence
ex b1 being Linear_Combination of V st Carrier b1 c= A
proof end;
end;

:: deftheorem Def6 defines Linear_Combination RLVECT_2:def 6 :
for V being non empty addLoopStr
for A being Subset of V
for b3 being Linear_Combination of V holds
( b3 is Linear_Combination of A iff Carrier b3 c= A );

theorem :: RLVECT_2:21
for V being RealLinearSpace
for A, B being Subset of V
for l being Linear_Combination of A st A c= B holds
l is Linear_Combination of B
proof end;

theorem Th22: :: RLVECT_2:22
for V being RealLinearSpace
for A being Subset of V holds ZeroLC V is Linear_Combination of A
proof end;

theorem Th23: :: RLVECT_2:23
for V being RealLinearSpace
for l being Linear_Combination of {} the carrier of V holds l = ZeroLC V
proof end;

definition
let V be RealLinearSpace;
let F be FinSequence of V;
let f be Function of the carrier of V,REAL;
func f (#) F -> FinSequence of the carrier of V means :Def7: :: RLVECT_2:def 7
( len it = len F & ( for i being Nat st i in dom it holds
it . i = (f . (F /. i)) * (F /. i) ) );
existence
ex b1 being FinSequence of the carrier of V st
( len b1 = len F & ( for i being Nat st i in dom b1 holds
b1 . i = (f . (F /. i)) * (F /. i) ) )
proof end;
uniqueness
for b1, b2 being FinSequence of the carrier of V st len b1 = len F & ( for i being Nat st i in dom b1 holds
b1 . i = (f . (F /. i)) * (F /. i) ) & len b2 = len F & ( for i being Nat st i in dom b2 holds
b2 . i = (f . (F /. i)) * (F /. i) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def7 defines (#) RLVECT_2:def 7 :
for V being RealLinearSpace
for F being FinSequence of V
for f being Function of the carrier of V,REAL
for b4 being FinSequence of the carrier of V holds
( b4 = f (#) F iff ( len b4 = len F & ( for i being Nat st i in dom b4 holds
b4 . i = (f . (F /. i)) * (F /. i) ) ) );

theorem Th24: :: RLVECT_2:24
for i being Nat
for V being RealLinearSpace
for v being VECTOR of V
for F being FinSequence of V
for f being Function of the carrier of V,REAL st i in dom F & v = F . i holds
(f (#) F) . i = (f . v) * v
proof end;

theorem :: RLVECT_2:25
for V being RealLinearSpace
for f being Function of the carrier of V,REAL holds f (#) (<*> the carrier of V) = <*> the carrier of V
proof end;

theorem Th26: :: RLVECT_2:26
for V being RealLinearSpace
for v being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v*> = <*((f . v) * v)*>
proof end;

theorem Th27: :: RLVECT_2:27
for V being RealLinearSpace
for v1, v2 being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v1,v2*> = <*((f . v1) * v1),((f . v2) * v2)*>
proof end;

theorem :: RLVECT_2:28
for V being RealLinearSpace
for v1, v2, v3 being VECTOR of V
for f being Function of the carrier of V,REAL holds f (#) <*v1,v2,v3*> = <*((f . v1) * v1),((f . v2) * v2),((f . v3) * v3)*>
proof end;

definition
let V be RealLinearSpace;
let L be Linear_Combination of V;
func Sum L -> Element of V means :Def8: :: RLVECT_2:def 8
ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & it = Sum (L (#) F) );
existence
ex b1 being Element of V ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b1 = Sum (L (#) F) )
proof end;
uniqueness
for b1, b2 being Element of V st ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b1 = Sum (L (#) F) ) & ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b2 = Sum (L (#) F) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def8 defines Sum RLVECT_2:def 8 :
for V being RealLinearSpace
for L being Linear_Combination of V
for b3 being Element of V holds
( b3 = Sum L iff ex F being FinSequence of V st
( F is one-to-one & rng F = Carrier L & b3 = Sum (L (#) F) ) );

Lm2: for V being RealLinearSpace holds Sum () = 0. V
proof end;

theorem :: RLVECT_2:29
for V being RealLinearSpace
for A being Subset of V holds
( ( A <> {} & A is linearly-closed ) iff for l being Linear_Combination of A holds Sum l in A )
proof end;

theorem :: RLVECT_2:30
for V being RealLinearSpace holds Sum () = 0. V by Lm2;

theorem :: RLVECT_2:31
for V being RealLinearSpace
for l being Linear_Combination of {} the carrier of V holds Sum l = 0. V
proof end;

theorem Th32: :: RLVECT_2:32
for V being RealLinearSpace
for v being VECTOR of V
for l being Linear_Combination of {v} holds Sum l = (l . v) * v
proof end;

theorem Th33: :: RLVECT_2:33
for V being RealLinearSpace
for v1, v2 being VECTOR of V st v1 <> v2 holds
for l being Linear_Combination of {v1,v2} holds Sum l = ((l . v1) * v1) + ((l . v2) * v2)
proof end;

theorem :: RLVECT_2:34
for V being RealLinearSpace
for L being Linear_Combination of V st Carrier L = {} holds
Sum L = 0. V
proof end;

theorem :: RLVECT_2:35
for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V st Carrier L = {v} holds
Sum L = (L . v) * v
proof end;

theorem :: RLVECT_2:36
for V being RealLinearSpace
for v1, v2 being VECTOR of V
for L being Linear_Combination of V st Carrier L = {v1,v2} & v1 <> v2 holds
Sum L = ((L . v1) * v1) + ((L . v2) * v2)
proof end;

definition
let V be non empty addLoopStr ;
let L1, L2 be Linear_Combination of V;
:: original: =
redefine pred L1 = L2 means :: RLVECT_2:def 9
for v being Element of V holds L1 . v = L2 . v;
compatibility
( L1 = L2 iff for v being Element of V holds L1 . v = L2 . v )
;
end;

:: deftheorem defines = RLVECT_2:def 9 :
for V being non empty addLoopStr
for L1, L2 being Linear_Combination of V holds
( L1 = L2 iff for v being Element of V holds L1 . v = L2 . v );

definition
let V be non empty addLoopStr ;
let L1, L2 be Linear_Combination of V;
:: original: +
redefine func L1 + L2 -> Linear_Combination of V means :Def10: :: RLVECT_2:def 10
for v being Element of V holds it . v = (L1 . v) + (L2 . v);
coherence
L1 + L2 is Linear_Combination of V
proof end;
compatibility
for b1 being Linear_Combination of V holds
( b1 = L1 + L2 iff for v being Element of V holds b1 . v = (L1 . v) + (L2 . v) )
proof end;
end;

:: deftheorem Def10 defines + RLVECT_2:def 10 :
for V being non empty addLoopStr
for L1, L2, b4 being Linear_Combination of V holds
( b4 = L1 + L2 iff for v being Element of V holds b4 . v = (L1 . v) + (L2 . v) );

theorem Th37: :: RLVECT_2:37
for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2)
proof end;

theorem Th38: :: RLVECT_2:38
for V being RealLinearSpace
for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 + L2 is Linear_Combination of A
proof end;

theorem :: RLVECT_2:39
for V being non empty addLoopStr
for L1, L2 being Linear_Combination of V holds L1 + L2 = L2 + L1 ;

theorem Th40: :: RLVECT_2:40
for V being RealLinearSpace
for L1, L2, L3 being Linear_Combination of V holds L1 + (L2 + L3) = (L1 + L2) + L3
proof end;

theorem Th41: :: RLVECT_2:41
for V being RealLinearSpace
for L being Linear_Combination of V holds
( L + () = L & () + L = L )
proof end;

definition
let V be RealLinearSpace;
let a be Real;
let L be Linear_Combination of V;
func a * L -> Linear_Combination of V means :Def11: :: RLVECT_2:def 11
for v being VECTOR of V holds it . v = a * (L . v);
existence
ex b1 being Linear_Combination of V st
for v being VECTOR of V holds b1 . v = a * (L . v)
proof end;
uniqueness
for b1, b2 being Linear_Combination of V st ( for v being VECTOR of V holds b1 . v = a * (L . v) ) & ( for v being VECTOR of V holds b2 . v = a * (L . v) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines * RLVECT_2:def 11 :
for V being RealLinearSpace
for a being Real
for L, b4 being Linear_Combination of V holds
( b4 = a * L iff for v being VECTOR of V holds b4 . v = a * (L . v) );

theorem Th42: :: RLVECT_2:42
for V being RealLinearSpace
for a being Real
for L being Linear_Combination of V st a <> 0 holds
Carrier (a * L) = Carrier L
proof end;

theorem Th43: :: RLVECT_2:43
for V being RealLinearSpace
for L being Linear_Combination of V holds 0 * L = ZeroLC V
proof end;

theorem Th44: :: RLVECT_2:44
for V being RealLinearSpace
for a being Real
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
a * L is Linear_Combination of A
proof end;

theorem Th45: :: RLVECT_2:45
for V being RealLinearSpace
for a, b being Real
for L being Linear_Combination of V holds (a + b) * L = (a * L) + (b * L)
proof end;

theorem Th46: :: RLVECT_2:46
for V being RealLinearSpace
for a being Real
for L1, L2 being Linear_Combination of V holds a * (L1 + L2) = (a * L1) + (a * L2)
proof end;

theorem Th47: :: RLVECT_2:47
for V being RealLinearSpace
for a, b being Real
for L being Linear_Combination of V holds a * (b * L) = (a * b) * L
proof end;

theorem Th48: :: RLVECT_2:48
for V being RealLinearSpace
for L being Linear_Combination of V holds 1 * L = L
proof end;

definition
let V be RealLinearSpace;
let L be Linear_Combination of V;
func - L -> Linear_Combination of V equals :: RLVECT_2:def 12
(- 1) * L;
correctness
coherence
(- 1) * L is Linear_Combination of V
;
;
end;

:: deftheorem defines - RLVECT_2:def 12 :
for V being RealLinearSpace
for L being Linear_Combination of V holds - L = (- 1) * L;

theorem Th49: :: RLVECT_2:49
for V being RealLinearSpace
for v being VECTOR of V
for L being Linear_Combination of V holds (- L) . v = - (L . v)
proof end;

theorem :: RLVECT_2:50
for V being RealLinearSpace
for L1, L2 being Linear_Combination of V st L1 + L2 = ZeroLC V holds
L2 = - L1
proof end;

theorem :: RLVECT_2:51
for V being RealLinearSpace
for L being Linear_Combination of V holds Carrier (- L) = Carrier L by Th42;

theorem :: RLVECT_2:52
for V being RealLinearSpace
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
- L is Linear_Combination of A by Th44;

theorem :: RLVECT_2:53
for V being RealLinearSpace
for L being Linear_Combination of V holds - (- L) = L
proof end;

definition
let V be RealLinearSpace;
let L1, L2 be Linear_Combination of V;
func L1 - L2 -> Linear_Combination of V equals :: RLVECT_2:def 13
L1 + (- L2);
correctness
coherence
L1 + (- L2) is Linear_Combination of V
;
;
end;

:: deftheorem defines - RLVECT_2:def 13 :
for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds L1 - L2 = L1 + (- L2);

theorem Th54: :: RLVECT_2:54
for V being RealLinearSpace
for v being VECTOR of V
for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)
proof end;

theorem :: RLVECT_2:55
for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds Carrier (L1 - L2) c= (Carrier L1) \/ (Carrier L2)
proof end;

theorem :: RLVECT_2:56
for V being RealLinearSpace
for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 - L2 is Linear_Combination of A
proof end;

theorem Th57: :: RLVECT_2:57
for V being RealLinearSpace
for L being Linear_Combination of V holds L - L = ZeroLC V
proof end;

definition
let V be RealLinearSpace;
func LinComb V -> set means :Def14: :: RLVECT_2:def 14
for x being set holds
( x in it iff x is Linear_Combination of V );
existence
ex b1 being set st
for x being set holds
( x in b1 iff x is Linear_Combination of V )
proof end;
uniqueness
for b1, b2 being set st ( for x being set holds
( x in b1 iff x is Linear_Combination of V ) ) & ( for x being set holds
( x in b2 iff x is Linear_Combination of V ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def14 defines LinComb RLVECT_2:def 14 :
for V being RealLinearSpace
for b2 being set holds
( b2 = LinComb V iff for x being set holds
( x in b2 iff x is Linear_Combination of V ) );

registration
let V be RealLinearSpace;
cluster LinComb V -> non empty ;
coherence
not LinComb V is empty
proof end;
end;

definition
let V be RealLinearSpace;
let e be Element of LinComb V;
func @ e -> Linear_Combination of V equals :: RLVECT_2:def 15
e;
coherence
e is Linear_Combination of V
by Def14;
end;

:: deftheorem defines @ RLVECT_2:def 15 :
for V being RealLinearSpace
for e being Element of LinComb V holds @ e = e;

definition
let V be RealLinearSpace;
let L be Linear_Combination of V;
func @ L -> Element of LinComb V equals :: RLVECT_2:def 16
L;
coherence
L is Element of LinComb V
by Def14;
end;

:: deftheorem defines @ RLVECT_2:def 16 :
for V being RealLinearSpace
for L being Linear_Combination of V holds @ L = L;

definition
let V be RealLinearSpace;
func LCAdd V -> BinOp of () means :Def17: :: RLVECT_2:def 17
for e1, e2 being Element of LinComb V holds it . (e1,e2) = (@ e1) + (@ e2);
existence
ex b1 being BinOp of () st
for e1, e2 being Element of LinComb V holds b1 . (e1,e2) = (@ e1) + (@ e2)
proof end;
uniqueness
for b1, b2 being BinOp of () st ( for e1, e2 being Element of LinComb V holds b1 . (e1,e2) = (@ e1) + (@ e2) ) & ( for e1, e2 being Element of LinComb V holds b2 . (e1,e2) = (@ e1) + (@ e2) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def17 defines LCAdd RLVECT_2:def 17 :
for V being RealLinearSpace
for b2 being BinOp of () holds
( b2 = LCAdd V iff for e1, e2 being Element of LinComb V holds b2 . (e1,e2) = (@ e1) + (@ e2) );

definition
let V be RealLinearSpace;
func LCMult V -> Function of [:REAL,():],() means :Def18: :: RLVECT_2:def 18
for a being Real
for e being Element of LinComb V holds it . [a,e] = a * (@ e);
existence
ex b1 being Function of [:REAL,():],() st
for a being Real
for e being Element of LinComb V holds b1 . [a,e] = a * (@ e)
proof end;
uniqueness
for b1, b2 being Function of [:REAL,():],() st ( for a being Real
for e being Element of LinComb V holds b1 . [a,e] = a * (@ e) ) & ( for a being Real
for e being Element of LinComb V holds b2 . [a,e] = a * (@ e) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def18 defines LCMult RLVECT_2:def 18 :
for V being RealLinearSpace
for b2 being Function of [:REAL,():],() holds
( b2 = LCMult V iff for a being Real
for e being Element of LinComb V holds b2 . [a,e] = a * (@ e) );

definition
let V be RealLinearSpace;
func LC_RLSpace V -> RLSStruct equals :: RLVECT_2:def 19
RLSStruct(# (),(@ ()),(),() #);
coherence
RLSStruct(# (),(@ ()),(),() #) is RLSStruct
;
end;

:: deftheorem defines LC_RLSpace RLVECT_2:def 19 :
for V being RealLinearSpace holds LC_RLSpace V = RLSStruct(# (),(@ ()),(),() #);

registration
let V be RealLinearSpace;
cluster LC_RLSpace V -> non empty strict ;
coherence
( LC_RLSpace V is strict & not LC_RLSpace V is empty )
;
end;

registration
let V be RealLinearSpace;
coherence
proof end;
end;

theorem :: RLVECT_2:58
for V being RealLinearSpace holds the carrier of () = LinComb V ;

theorem :: RLVECT_2:59
for V being RealLinearSpace holds 0. () = ZeroLC V ;

theorem :: RLVECT_2:60
for V being RealLinearSpace holds the addF of () = LCAdd V ;

theorem :: RLVECT_2:61
for V being RealLinearSpace holds the Mult of () = LCMult V ;

theorem Th62: :: RLVECT_2:62
for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds (vector ((),L1)) + (vector ((),L2)) = L1 + L2
proof end;

theorem Th63: :: RLVECT_2:63
for V being RealLinearSpace
for a being Real
for L being Linear_Combination of V holds a * (vector ((),L)) = a * L
proof end;

theorem Th64: :: RLVECT_2:64
for V being RealLinearSpace
for L being Linear_Combination of V holds - (vector ((),L)) = - L
proof end;

theorem :: RLVECT_2:65
for V being RealLinearSpace
for L1, L2 being Linear_Combination of V holds (vector ((),L1)) - (vector ((),L2)) = L1 - L2
proof end;

definition
let V be RealLinearSpace;
let A be Subset of V;
func LC_RLSpace A -> strict Subspace of LC_RLSpace V means :: RLVECT_2:def 20
the carrier of it = { l where l is Linear_Combination of A : verum } ;
existence
ex b1 being strict Subspace of LC_RLSpace V st the carrier of b1 = { l where l is Linear_Combination of A : verum }
proof end;
uniqueness
for b1, b2 being strict Subspace of LC_RLSpace V st the carrier of b1 = { l where l is Linear_Combination of A : verum } & the carrier of b2 = { l where l is Linear_Combination of A : verum } holds
b1 = b2
by RLSUB_1:30;
end;

:: deftheorem defines LC_RLSpace RLVECT_2:def 20 :
for V being RealLinearSpace
for A being Subset of V
for b3 being strict Subspace of LC_RLSpace V holds
( b3 = LC_RLSpace A iff the carrier of b3 = { l where l is Linear_Combination of A : verum } );

theorem Th66: :: RLVECT_2:66
for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for F, G being FinSequence of V st len F = len G & ( for k being Nat
for v being Element of V st k in dom F & v = G . k holds
F . k = a * v ) holds
Sum F = a * (Sum G)
proof end;

theorem :: RLVECT_2:67
for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for F, G being FinSequence of V st len F = len G & ( for k being Nat st k in dom F holds
G . k = a * (F /. k) ) holds
Sum G = a * (Sum F)
proof end;

theorem :: RLVECT_2:68
for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for V being non empty right_complementable Abelian add-associative right_zeroed ModuleStr over R
for F, G, H being FinSequence of V st len F = len G & len F = len H & ( for k being Nat st k in dom F holds
H . k = (F /. k) - (G /. k) ) holds
Sum H = (Sum F) - (Sum G)
proof end;

theorem :: RLVECT_2:69
for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R holds a * (Sum (<*> the carrier of V)) = 0. V
proof end;

theorem :: RLVECT_2:70
for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for v, u being Element of V holds a * (Sum <*v,u*>) = (a * v) + (a * u)
proof end;

theorem :: RLVECT_2:71
for R being non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr
for a being Element of R
for V being non empty right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital ModuleStr over R
for v, u, w being Element of V holds a * (Sum <*v,u,w*>) = ((a * v) + (a * u)) + (a * w)
proof end;