
theorem
  for L be Abelian add-associative right_zeroed right_complementable
  well-unital commutative distributive non empty doubleLoopStr for p be
  Polynomial of L holds Subst(p,0_.(L)) = <%p.0%>
proof
  let L be Abelian add-associative right_zeroed right_complementable
  well-unital commutative distributive non empty doubleLoopStr;
  let p be Polynomial of L;
  consider F be FinSequence of the carrier of Polynom-Ring L such that
A1: Subst(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)*((0_.(L))
  `^(n-'1)) by Def6;
  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 n be Element of NAT;
      assume that
A5:   n in dom F and
A6:   n <> 1;
      n >= 1 by A5,FINSEQ_3:25;
      then
A7:   n > 0+1 by A6,XXREAL_0:1;
      then n >= 1+1 by NAT_1:13;
      then
A8:   n-2 >= 1+1-2 by XREAL_1:9;
      n-1 >= 0 by A7;
      then
A9:   n-'1 = n-(1+1)+1 by XREAL_0:def 2
        .= n-'2+1 by A8,XREAL_0:def 2;
      thus F/.n = F.n by A5,PARTFUN1:def 6
        .= p.(n-'1)*((0_.(L))`^(n-'1)) by A3,A5
        .= p.(n-'1)*(0_.(L)) by A9,Th20
        .= 0_.(L) by Th28
        .= 0.(Polynom-Ring L) by POLYNOM3:def 10;
    end;
    hence Subst(p,0_.(L)) = F/.1 by A1,A4,POLYNOM2:3
      .= F.1 by A4,PARTFUN1:def 6
      .= p.(1-'1)*((0_.(L))`^(1-'1)) by A3,A4
      .= p.(1-'1)*((0_.(L))`^0) by XREAL_1:232
      .= p.0*((0_.(L))`^0) by XREAL_1:232
      .= p.0*(1_.(L)) by Th15
      .= <%p.0%> by Th29;
  end;
  suppose
    len F = 0;
    then
A10: p = 0_.(L) by A2,POLYNOM4:5;
    hence Subst(p,0_.(L)) = 0_.(L) by Th49
      .= <%0.L%> by Th34
      .= <%p.0%> by A10,FUNCOP_1:7;
  end;
end;
