
theorem Th35:
for L being Field
for p being rational_function of L
for x being Element of L st eval(p`2,x) <> 0.L
holds eval(NormRationalFunction p,x) = eval(p,x)
proof
let L be Field; let p be rational_function of L; let x be Element of L;
assume A1: eval(p`2,x) <> 0.L;
set np = NormRationalFunction p;
A2: 1.L / LC(p`2) <> 0.L;
thus eval(np,x)
   = eval(1.L / LC(p`2) * p`1,x) * eval(np`2,x)"
  .= eval(1.L / LC(p`2) * p`1,x) * eval(1.L / LC(p`2) * p`2,x)"
  .= (1.L / LC(p`2) * eval(p`1,x)) * eval(1.L / LC(p`2) * p`2,x)"
     by POLYNOM5:30
  .= (1.L / LC(p`2) * eval(p`1,x)) * (1.L / LC(p`2) * eval(p`2,x))"
     by POLYNOM5:30
  .= (1.L / LC(p`2) * eval(p`1,x)) * ((1.L / LC(p`2))" * eval(p`2,x)")
     by A1,VECTSP_2:11
  .= eval(p`1,x) * (1.L / LC(p`2) * ((1.L / LC(p`2))" * eval(p`2,x)"))
     by GROUP_1:def 3
  .= eval(p`1,x) * ((1.L / LC(p`2) * (1.L / LC(p`2))") * eval(p`2,x)")
     by GROUP_1:def 3
  .= eval(p`1,x) * (1.L * eval(p`2,x)") by A2,VECTSP_1:def 10
  .= eval(p,x);
end;
