:: HILBASIS semantic presentation

for A, B being FinSequence
for f being Function st (rng A) \/ (rng B) c= dom f holds
ex fA, fB being FinSequence st
( fA = f * A & fB = f * B & f * (A ^ B) = fA ^ fB )

proof end;

theorem Th2: :: HILBASIS:2

for b being bag of 0 holds decomp b = <*<*{} ,{} *>*>

proof end;

theorem Th3: :: HILBASIS:3

for i, j being Nat
for b being bag of j st i <= j holds
b | i is Element of Bags i

proof end;

theorem Th4: :: HILBASIS:4

for i, j being set
for b1, b2 being bag of j
for b1', b2' being bag of i st b1' = b1 | i & b2' = b2 | i & b1 divides b2 holds
b1' divides b2'

proof end;

theorem Th5: :: HILBASIS:5

for i, j being set
for b1, b2 being bag of j
for b1', b2' being bag of i st b1' = b1 | i & b2' = b2 | i holds
( (b1 -' b2) | i = b1' -' b2' & (b1 + b2) | i = b1' + b2' )

proof end;

definition

let n, k be Nat;
let b be bag of n;
func b bag_extend k -> Element of Bags (n + 1) means :Def1: :: HILBASIS:def 1
( it | n = b & it . n = k );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def1 defines bag_extend HILBASIS:def 1 :

theorem Th6: :: HILBASIS:6

for n being Nat holds EmptyBag (n + 1) = (EmptyBag n) bag_extend 0

proof end;

theorem Th7: :: HILBASIS:7

for n being Ordinal
for b, b1 being bag of n holds
( b1 in rng (divisors b) iff b1 divides b )

proof end;

definition

let X be set ;
let x be Element of X;
func UnitBag x -> Element of Bags X equals :: HILBASIS:def 2
(EmptyBag X) +* x,1;
coherence by POLYNOM1:def 14;

end;

:: deftheorem defines UnitBag HILBASIS:def 2 :

theorem Th8: :: HILBASIS:8

for X being non empty set
for x being Element of X holds support (UnitBag x) = {x}

proof end;

theorem Th9: :: HILBASIS:9

for X being non empty set
for x being Element of X holds
( (UnitBag x) . x = 1 & ( for y being Element of X st x <> y holds
(UnitBag x) . y = 0 ) )

proof end;

theorem Th10: :: HILBASIS:10

for X being non empty set
for x1, x2 being Element of X st UnitBag x1 = UnitBag x2 holds
x1 = x2

proof end;

theorem Th11: :: HILBASIS:11

for X being non empty Ordinal
for x being Element of X
for L being non empty unital non trivial doubleLoopStr
for e being Function of X,L holds eval (UnitBag x),e = e . x

proof end;

definition

let X be set ;
let x be Element of X;
let L be non empty unital multLoopStr_0 ;
func 1_1 x,L -> Series of X,L equals :: HILBASIS:def 3
(0_ X,L) +* (UnitBag x),(1. L);
coherence ;

end;

:: deftheorem defines 1_1 HILBASIS:def 3 :

theorem Th12: :: HILBASIS:12

for X being set
for L being non empty unital non trivial doubleLoopStr
for x being Element of X holds
( (1_1 x,L) . (UnitBag x) = 1. L & ( for b being bag of X st b <> UnitBag x holds
(1_1 x,L) . b = 0. L ) )

proof end;

theorem Th13: :: HILBASIS:13

for X being set
for x being Element of X
for L being non empty unital add-associative right_zeroed right_complementable right-distributive non trivial doubleLoopStr holds Support (1_1 x,L) = {(UnitBag x)}

proof end;

registration

let X be Ordinal;
let x be Element of X;
let L be non empty unital add-associative right_zeroed right_complementable right-distributive non trivial doubleLoopStr ;
cluster 1_1 x,L -> finite-Support ;
coherence

proof end;

end;

theorem Th14: :: HILBASIS:14

for L being non empty unital add-associative right_zeroed right_complementable right-distributive non trivial doubleLoopStr
for X being non empty set
for x1, x2 being Element of X st 1_1 x1,L = 1_1 x2,L holds
x1 = x2

proof end;

theorem Th15: :: HILBASIS:15

for L being non empty add-associative right_zeroed right_complementable distributive doubleLoopStr
for x being Element of (Polynom-Ring L)
for p being sequence of L st x = p holds
- x = - p

proof end;

theorem Th16: :: HILBASIS:16

for L being non empty add-associative right_zeroed right_complementable distributive doubleLoopStr
for x, y being Element of (Polynom-Ring L)
for p, q being sequence of L st x = p & y = q holds
x - y = p - q

proof end;

definition

let L be non empty unital add-associative right_zeroed right_complementable distributive doubleLoopStr ;
let I be non empty Subset of (Polynom-Ring L);
func minlen I -> non empty Subset of I equals :: HILBASIS:def 4
{ x where x is Element of I : for x', y' being Polynomial of L st x' = x & y' in I holds
len x' <= len y' } ;
coherence

proof end;

end;

:: deftheorem defines minlen HILBASIS:def 4 :

theorem Th17: :: HILBASIS:17

for L being non empty unital add-associative right_zeroed right_complementable distributive doubleLoopStr
for I being non empty Subset of (Polynom-Ring L)
for i1, i2 being Polynomial of L st i1 in minlen I & i2 in I holds
( i1 in I & len i1 <= len i2 )

proof end;

definition

let L be non empty unital add-associative right_zeroed right_complementable distributive doubleLoopStr ;
let n be Nat;
let a be Element of L;
func monomial a,n -> Polynomial of L means :Def5: :: HILBASIS:def 5
for x being Nat holds
( ( x = n implies it . x = a ) & ( x <> n implies it . x = 0. L ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines monomial HILBASIS:def 5 :

theorem Th18: :: HILBASIS:18

for L being non empty unital add-associative right_zeroed right_complementable distributive doubleLoopStr
for n being Nat
for a being Element of L holds
( ( a <> 0. L implies len (monomial a,n) = n + 1 ) & ( a = 0. L implies len (monomial a,n) = 0 ) & len (monomial a,n) <= n + 1 )

proof end;

theorem Th19: :: HILBASIS:19

for L being non empty unital add-associative right_zeroed right_complementable distributive doubleLoopStr
for n, x being Nat
for a being Element of L
for p being Polynomial of L holds ((monomial a,n) *' p) . (x + n) = a * (p . x)

proof end;

theorem Th20: :: HILBASIS:20

proof end;

theorem Th21: :: HILBASIS:21

for L being non empty unital add-associative right_zeroed right_complementable distributive doubleLoopStr
for p, q being Polynomial of L holds len (p *' q) <= ((len p) + (len q)) -' 1

proof end;

theorem Th22: :: HILBASIS:22

for R, S being non empty doubleLoopStr
for I being Ideal of R
for P being Function of R,S st P is RingIsomorphism holds
P .: I is Ideal of S

proof end;

theorem Th23: :: HILBASIS:23

for R, S being non empty add-associative right_zeroed right_complementable doubleLoopStr
for f being Function of R,S st f is RingHomomorphism holds
f . (0. R) = 0. S

proof end;

theorem Th24: :: HILBASIS:24

for R, S being non empty add-associative right_zeroed right_complementable doubleLoopStr
for F being non empty Subset of R
for G being non empty Subset of S
for P being Function of R,S
for lc being LinearCombination of F
for LC being LinearCombination of G
for E being FinSequence of [:the carrier of R,the carrier of R,the carrier of R:] st P is RingHomomorphism & len lc = len LC & E represents lc & ( for i being set st i in dom LC holds
LC . i = ((P . ((E /. i) `1 )) * (P . ((E /. i) `2 ))) * (P . ((E /. i) `3 )) ) holds
P . (Sum lc) = Sum LC

proof end;

theorem Th25: :: HILBASIS:25

for R, S being non empty doubleLoopStr
for P being Function of R,S st P is RingIsomorphism holds
ex P1 being Function of S,R st
( P1 is RingIsomorphism & P1 = P " )

proof end;

theorem Th26: :: HILBASIS:26