
theorem Th20:
for L being add-associative right_zeroed right_complementable
            well-unital commutative associative distributive
            almost_left_invertible non trivial doubleLoopStr
for p being rational_function of L
for a being non zero Element of L
holds [a * p`1, a * p`2] is irreducible iff p is irreducible
proof
let L be add-associative right_zeroed right_complementable
         well-unital commutative associative distributive
         almost_left_invertible non trivial doubleLoopStr;
let p be rational_function of L;
let a be non zero Element of L;
set ap = [a * p`1, a * p`2];
A1: now assume A2: p is irreducible;
   assume ap is reducible;
   then ap`1, ap`2 have_common_roots;
   then consider x being Element of L such that
   A3: x is_a_common_root_of ap`1, ap`2;
   x is_a_root_of ap`1 & x is_a_root_of ap`2 by A3;
   then A4: eval(ap`1,x) = 0.L & eval(ap`2,x) = 0.L by POLYNOM5:def 7;
   then eval(a * p`1,x) = 0.L;
   then a * eval(p`1,x) = 0.L by POLYNOM5:30;
   then eval(p`1,x) = 0.L by VECTSP_2:def 1;
   then A5: x is_a_root_of p`1 by POLYNOM5:def 7;
   eval(a * p`2,x) = 0.L by A4;
   then a * eval(p`2,x) = 0.L by POLYNOM5:30;
   then eval(p`2,x) = 0.L by VECTSP_2:def 1;
   then x is_a_root_of p`2 by POLYNOM5:def 7;
   then x is_a_common_root_of p`1,p`2 by A5;
   then p`1,p`2 have_common_roots;
   hence [a * p`1, a * p`2] is irreducible by A2;
   end;
now assume A6: ap is irreducible;
   assume p is reducible;
   then p`1, p`2 have_common_roots;
   then consider x being Element of L such that
   A7: x is_a_common_root_of p`1, p`2;
   x is_a_root_of p`1 & x is_a_root_of p`2 by A7;
   then A8: eval(p`1,x) = 0.L & eval(p`2,x) = 0.L by POLYNOM5:def 7;
   then a * eval(p`1,x) = 0.L;
   then eval(a * p`1,x) = 0.L by POLYNOM5:30;
   then eval(ap`1,x) = 0.L;
   then A9: x is_a_root_of ap`1 by POLYNOM5:def 7;
   a * eval(p`2,x) = 0.L by A8;
   then eval(a * p`2,x) = 0.L by POLYNOM5:30;
   then eval(ap`2,x) = 0.L;
   then x is_a_root_of ap`2 by POLYNOM5:def 7;
   then x is_a_common_root_of ap`1,ap`2 by A9;
   then ap`1,ap`2 have_common_roots;
   hence p is irreducible by A6;
   end;
hence thesis by A1;
end;
