
theorem
for L being Abelian add-associative right_zeroed right_complementable
         well-unital associative distributive commutative
         almost_left_invertible domRing-like non trivial doubleLoopStr
for z being rational_function of L
holds NF (NF z) = NF z
proof
let L be Abelian add-associative right_zeroed right_complementable
         well-unital associative distributive commutative
         almost_left_invertible domRing-like non trivial doubleLoopStr;
let z be rational_function of L;
set nfz = NF z;
per cases;
suppose z is zero;
 then A1: NF z = 0._(L) by Def17;
 thus thesis by A1,Def17;
 end;
suppose A2: z is non zero;
 A3: 1.L <> 0.L;
 A4: NF nfz = NormRatF nfz by A2,Lm4
               .= [(1.L/LC(nfz`2)) * nfz`1, (1.L/LC(nfz`2)) * nfz`2];
 nfz`2 is normalized by Def11;
 then A5: LC(nfz`2) = 1.L;
 A6: 1.L / LC(nfz`2) = (LC(nfz`2))"*LC(nfz`2) by A5
       .= 1.L by VECTSP_1:def 10,A3;
 then NF nfz = [nfz`1, (1.L / LC(nfz`2)) * nfz`2] by A4,POLYNOM5:27
            .= [nfz`1, nfz`2] by A6,POLYNOM5:27
            .= nfz by Th19;
 hence thesis;
 end;
end;
