
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 non zero rational_function of L
holds z is irreducible iff degree z = max( degree(z`1), degree(z`2) )
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 non zero rational_function of L;
set p1 = z`1, p2 = z`2;
A1: now assume z is irreducible;
   then consider a being Element of L such that
   A2: a <> 0.L & [a * (z`1), a * (z`2)] = NF(z) by Th28;
   A3: degree(a * (z`1)) = degree(z`1) by A2,POLYNOM5:25;
   A4: degree(a * (z`2)) = degree(z`2) by A2,POLYNOM5:25;
   degree((NF(z))`1) = degree(z`1) by A3,A2;
   hence degree z = max(degree(z`1),degree(z`2)) by A4,A2;
   end;
now assume A5: degree z = max( degree(z`1), degree(z`2));
   now assume not(z is irreducible);
     then p1, p2 have_a_common_root;
     then consider x being Element of L such that
     A6: x is_a_common_root_of p1,p2;
     A7: x is_a_root_of p1 & x is_a_root_of p2 by A6;
     then consider q1 being Polynomial of L such that
     A8: p1 = rpoly(1,x) *' q1 by HURWITZ:33;
     consider q2 being Polynomial of L such that
     A9: p2 = rpoly(1,x) *' q2 by A7,HURWITZ:33;
     q2 <> 0_.(L) by A9,POLYNOM3:34;
     then reconsider q2 as non zero Polynomial of L by UPROOTS:def 5;
     set zz = [q1,q2];
     A10: zz`1 = q1 & zz`2 = q2;
     z = [rpoly(1,x) *' zz`1,rpoly(1,x) *' zz`2] by Th19,A8,A9;
     then NF(z) = NF(zz) by Th26;
     then degree z =  degree zz;
     then A11: degree(z) <= max(degree(q1),degree(q2)) by Th29,A10;
     now per cases;
     suppose A12: p1 = 0_.(L);
       A13: q1 = 0_.(L) by A12,A8,Lm5;
       deg(rpoly(1,x)*'q2) + 0
           = deg(rpoly(1,x)) + deg(q2) by HURWITZ:23
          .= 1 + deg(q2) by HURWITZ:27;
       then A14: deg(q2) < deg(p2) by A9,XREAL_1:8;
       deg(p1) <= deg(p2) by A12,HURWITZ:20;
       then A15: max(deg(p1),deg(p2)) = deg(p2) by XXREAL_0:def 10;
       deg(q1) <= deg(q2) by A13,HURWITZ:20;
       hence contradiction by A15,A14,A5,A11,XXREAL_0:def 10;
     end;
     suppose p1 <> 0_.(L);
       then reconsider p1 as non zero Polynomial of L by UPROOTS:def 5;
       now assume q1 = 0_.(L);
          then p1 = 0_.(L) by A8,POLYNOM3:34;
          hence contradiction;
          end;
       then reconsider q1 as non zero Polynomial of L by UPROOTS:def 5;
       deg(p1) + 0 = deg(rpoly(1,x)) + deg(q1) by A8,HURWITZ:23
                 .= 1 + deg(q1) by HURWITZ:27;
       then A16: deg(q1) < deg(p1) by XREAL_1:8;
       deg(p2) + 0 = deg(rpoly(1,x)) + deg(q2) by A9,HURWITZ:23
                  .= 1 + deg(q2) by HURWITZ:27;
       then deg(q2) < deg(p2) by XREAL_1:8;
       hence contradiction by A5,A11,A16,XXREAL_0:27;
       end;
     end;
     hence contradiction;
     end;
   hence z is irreducible;
   end;
hence thesis by A1;
end;
