let n be Ordinal; for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g in P -Ideal
let T be connected TermOrder of n; for L being non trivial right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g in P -Ideal
let L be non trivial right_complementable almost_left_invertible well-unital distributive Abelian add-associative right_zeroed associative commutative doubleLoopStr ; for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g in P -Ideal
let P be Subset of (Polynom-Ring (n,L)); for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g in P -Ideal
let f, g be Polynomial of n,L; ( PolyRedRel (P,T) reduces f,g implies f - g in P -Ideal )
reconsider f9 = f, g9 = g as Element of (Polynom-Ring (n,L)) by POLYNOM1:def 11;
reconsider f9 = f9, g9 = g9 as Element of (Polynom-Ring (n,L)) ;
set R = Polynom-Ring (n,L);
reconsider x = - g as Element of (Polynom-Ring (n,L)) by POLYNOM1:def 11;
reconsider x = x as Element of (Polynom-Ring (n,L)) ;
x + g9 =
(- g) + g
by POLYNOM1:def 11
.=
0_ (n,L)
by Th3
.=
0. (Polynom-Ring (n,L))
by POLYNOM1:def 11
;
then A1:
- g9 = - g
by RLVECT_1:6;
assume
PolyRedRel (P,T) reduces f,g
; f - g in P -Ideal
then
f,g are_convertible_wrt PolyRedRel (P,T)
by REWRITE1:25;
then A2:
f9,g9 are_congruent_mod P -Ideal
by Th57;
f - g =
f + (- g)
by POLYNOM1:def 7
.=
f9 + (- g9)
by A1, POLYNOM1:def 11
.=
f9 - g9
;
hence
f - g in P -Ideal
by A2; verum