
theorem Th30:
  for L be add-associative right_zeroed right_complementable
  well-unital distributive commutative associative almost_left_invertible non
  empty doubleLoopStr for p be Polynomial of L for v,x be Element of L holds
  eval(v*p,x) = v*eval(p,x)
proof
  let L be add-associative right_zeroed right_complementable well-unital
  distributive commutative associative almost_left_invertible non empty
  doubleLoopStr;
  let p be Polynomial of L;
  let v,x be Element of L;
  consider F1 be FinSequence of the carrier of L such that
A1: eval(p,x) = Sum F1 and
A2: len F1 = len p and
A3: for n be Element of NAT st n in dom F1 holds F1.n = p.(n-'1) * (
  power L).(x,n-'1) by POLYNOM4:def 2;
  consider F2 be FinSequence of the carrier of L such that
A4: eval(v*p,x) = Sum F2 and
A5: len F2 = len (v*p) and
A6: for n be Element of NAT st n in dom F2 holds F2.n = (v*p).(n-'1) * (
  power L).(x,n-'1) by POLYNOM4:def 2;
  per cases;
  suppose
    v <> 0.L;
    then len F1 = len F2 by A2,A5,Th25;
    then
A7: dom F1 = dom F2 by FINSEQ_3:29;
    now
      let i be object;
      assume
A8:   i in dom F1;
      then reconsider i1=i as Element of NAT;
A9:   p.(i1-'1) * (power L).(x,i1-'1) = F1.i by A3,A8
        .= F1/.i by A8,PARTFUN1:def 6;
      thus F2/.i = F2.i by A7,A8,PARTFUN1:def 6
        .= (v*p).(i1-'1) * (power L).(x,i1-'1) by A6,A7,A8
        .= v*p.(i1-'1) * (power L).(x,i1-'1) by Def4
        .= v*(F1/.i) by A9,GROUP_1:def 3;
    end;
    then F2 = v*F1 by A7,POLYNOM1:def 1;
    hence thesis by A1,A4,POLYNOM1:12;
  end;
  suppose
A10: v = 0.L;
    hence eval(v*p,x) = eval(0_.(L),x) by Th26
      .= 0.L by POLYNOM4:17
      .= v*eval(p,x) by A10;
  end;
end;
