
theorem Th31:
  for L be add-associative right_zeroed right_complementable
  right-distributive unital non empty doubleLoopStr for p be Polynomial of L
  holds eval(p,0.L) = p.0
proof
  let L be add-associative right_zeroed right_complementable
  right-distributive unital non empty doubleLoopStr;
  let p be Polynomial of L;
  consider F be FinSequence of the carrier of L such that
A1: eval(p,0.L) = Sum F and
A2: len F = len p and
A3: for n be Element of NAT st n in dom F holds F.n = p.(n-'1) * (power
  L).(0.L,n-'1) by POLYNOM4:def 2;
  per cases;
  suppose
    len F > 0;
    then 0+1 <= len F by NAT_1:13;
    then
A4: 1 in dom F by FINSEQ_3:25;
    now
      let i be Element of NAT;
      assume that
A5:   i in dom F and
A6:   i <> 1;
      0+1 <= i by A5,FINSEQ_3:25;
      then i > 0+1 by A6,XXREAL_0:1;
      then i-1 > 0 by XREAL_1:20;
      then
A7:   i-'1 > 0 by XREAL_0:def 2;
      thus F/.i = F.i by A5,PARTFUN1:def 6
        .= p.(i-'1) * (power L).(0.L,i-'1) by A3,A5
        .= p.(i-'1) * 0.L by A7,VECTSP_1:36
        .= 0.L;
    end;
    hence eval(p,0.L) = F/.1 by A1,A4,POLYNOM2:3
      .= F.1 by A4,PARTFUN1:def 6
      .= p.(1-'1) * (power L).(0.L,1-'1) by A3,A4
      .= p.(1-'1) * (power L).(0.L,0) by XREAL_1:232
      .= p.(1-'1) * 1_L by GROUP_1:def 7
      .= p.(1-'1) by GROUP_1:def 4
      .= p.0 by XREAL_1:232;
  end;
  suppose
    len F = 0;
    then
A8: p = 0_.(L) by A2,POLYNOM4:5;
    hence eval(p,0.L) = 0.L by POLYNOM4:17
      .= p.0 by A8,FUNCOP_1:7;
  end;
end;
