begin
:: deftheorem POLYNOM7:def 1 :
canceled;
:: deftheorem Def2 defines non-zero POLYNOM7:def 2 :
theorem Th1:
theorem
:: deftheorem Def3 defines univariate POLYNOM7:def 3 :
begin
theorem
Lm1:
for L being non empty doubleLoopStr
for p being Polynomial of {} ,L ex a being Element of L st p = {(EmptyBag {} )} --> a
theorem
theorem
begin
:: deftheorem Def4 defines monomial-like POLYNOM7:def 4 :
theorem Th6:
:: deftheorem defines Monom POLYNOM7:def 5 :
:: deftheorem Def6 defines term POLYNOM7:def 6 :
:: deftheorem defines coefficient POLYNOM7:def 7 :
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem
theorem Th12:
theorem
begin
:: deftheorem Def8 defines Constant POLYNOM7:def 8 :
theorem Th14:
Lm2:
for X being set
for L being non empty ZeroStr
for c being ConstPoly of X,L holds
( term c = EmptyBag X & coefficient c = c . (EmptyBag X) )
theorem Th15:
theorem
:: deftheorem defines | POLYNOM7:def 9 :
Lm3:
for X being set
for L being non empty ZeroStr holds (0. L) | X,L = 0_ X,L
theorem
theorem Th18:
theorem
theorem
theorem
theorem
theorem Th23:
theorem Th24:
theorem Th25:
begin
:: deftheorem Def10 defines * POLYNOM7:def 10 :
:: deftheorem Def11 defines * POLYNOM7:def 11 :
theorem
theorem Th27:
theorem Th28:
Lm4:
for n being Ordinal
for L being non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr
for p being Polynomial of n,L
for a being Element of L
for x being Function of n,L holds eval ((a | n,L) *' p),x = a * (eval p,x)
Lm5:
for n being Ordinal
for L being non trivial right_complementable Abelian add-associative right_zeroed associative commutative well-unital distributive left_zeroed doubleLoopStr
for p being Polynomial of n,L
for a being Element of L
for x being Function of n,L holds eval (p *' (a | n,L)),x = (eval p,x) * a
theorem
theorem
theorem
theorem
theorem
theorem