
theorem Th28:
for L being Abelian add-associative right_zeroed right_complementable
         well-unital associative distributive commutative
         almost_left_invertible domRing-like non degenerated doubleLoopStr
for z being non zero rational_function of L
holds z is irreducible iff
ex a being Element of L st a <> 0.L & [a * (z`1), a * (z`2)] = NF(z)
proof
let L be Abelian add-associative right_zeroed right_complementable
         well-unital associative distributive commutative
         almost_left_invertible domRing-like non degenerated doubleLoopStr;
let z be non zero rational_function of L;
set q = z`2;
set a = (LC(z`2))";
now assume A1: a = 0.L;
    then A2: a * (LC q) = 0.L;
    a <> 1.L by A1;
    hence contradiction by A2,VECTSP_1:def 10;
    end;
then reconsider a as non zero Element of L by STRUCT_0:def 12;
reconsider x = [a * (z`1), a * (z`2)] as rational_function of L;
A3: now assume z is irreducible;
   then NF z = NormRatF z by Lm4
              .= [(1.L / LC(z`2)) * z`1, (1.L / LC(z`2)) * z`2];
   hence ex a being Element of L st a <> 0.L &
                          [a * (z`1), a * (z`2)] = NF(z);
   end;
now assume
  ex a being Element of L st a <> 0.L & [a * (z`1), a * (z`2)] = NF(z);
  then consider a being Element of L such that
  A4: a <> 0.L & [a * (z`1), a * (z`2)] = NF(z);
  reconsider x = [a * (z`1), a * (z`2)] as rational_function of L by A4;
  now assume z`1, z`2 have_a_common_root;
    then consider u being Element of L such that
    A5: u is_a_common_root_of z`1, z`2;
    u is_a_root_of z`1 & u is_a_root_of z`2 by A5;
    then A6: eval(z`1,u) = 0.L & eval(z`2,u) = 0.L by POLYNOM5:def 7;
    eval(x`1,u) = a * eval(z`1,u) by POLYNOM5:30;
    then eval(x`1,u) = 0.L by A6;
    then A7: u is_a_root_of x`1 by POLYNOM5:def 7;
    eval(x`2,u) = a * eval(z`2,u) by POLYNOM5:30;
    then eval(x`2,u) = 0.L by A6;
    then u is_a_root_of x`2 by POLYNOM5:def 7;
    then u is_a_common_root_of x`1, x`2 by A7;
    then x`1,x`2 have_a_common_root;
    hence contradiction by Def10,A4;
    end;
  hence z is irreducible;
  end;
hence thesis by A3;
end;
