let n be Ordinal; :: thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed 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, 0_ n,L holds
f,g are_convergent_wrt PolyRedRel P,T
let T be connected TermOrder of n; :: thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed 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, 0_ n,L holds
f,g are_convergent_wrt PolyRedRel P,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for P being Subset of (Polynom-Ring n,L)
for f, g being Polynomial of n,L st PolyRedRel P,T reduces f - g, 0_ n,L holds
f,g are_convergent_wrt PolyRedRel P,T
let P be Subset of (Polynom-Ring n,L); :: thesis: for f, g being Polynomial of n,L st PolyRedRel P,T reduces f - g, 0_ n,L holds
f,g are_convergent_wrt PolyRedRel P,T
let f, g be Polynomial of n,L; :: thesis: ( PolyRedRel P,T reduces f - g, 0_ n,L implies f,g are_convergent_wrt PolyRedRel P,T )
assume PolyRedRel P,T reduces f - g, 0_ n,L ; :: thesis: f,g are_convergent_wrt PolyRedRel P,T
then consider f1, g1 being Polynomial of n,L such that
A1: f1 - g1 = 0_ n,L and
A2: ( PolyRedRel P,T reduces f,f1 & PolyRedRel P,T reduces g,g1 ) by Th49;
g1 = (f1 - g1) + g1 by A1, Th2
.= (f1 + (- g1)) + g1 by POLYNOM1:def 23
.= f1 + ((- g1) + g1) by POLYNOM1:80
.= f1 + (0_ n,L) by Th3
.= f1 by POLYNOM1:82 ;
hence f,g are_convergent_wrt PolyRedRel P,T by A2, REWRITE1:def 7; :: thesis: verum