
theorem Th20:
  for L be right_zeroed add-associative right_complementable
  well-unital distributive non empty doubleLoopStr, n, x be Element of NAT, a
be Element of L, p be Polynomial of L holds (p*'monomial(a,n)).(x+n) = (p.x) *
  a
proof
  let L be right_zeroed add-associative right_complementable well-unital
distributive non empty doubleLoopStr, n,x be Element of NAT, a be Element of
  L, p be Polynomial of L;
  consider r being FinSequence of the carrier of L such that
A1: len r = x+n+1 and
A2: (p*'monomial(a,n)).(x+n) = Sum r and
A3: for k be Element of NAT st k in dom r holds r.k = p.(k-'1) *
  monomial(a,n).(x+n+1-'k) by POLYNOM3:def 9;
  len r = x+(n+1) by A1;
  then consider b,c being FinSequence of the carrier of L such that
A4: len b = x and
A5: len c = n+1 and
A6: r = b^c by FINSEQ_2:23;
  consider d,e being FinSequence of the carrier of L such that
A7: len d = 1 and
A8: len e = n and
A9: c = d^e by A5,FINSEQ_2:23;
A10: dom d c= dom c by A9,FINSEQ_1:26;
  now
    let i be Element of NAT;
A11: n > n-1 by XREAL_1:146;
    assume
A12: i in dom e;
    then
A13: 1+i in dom c by A7,A9,FINSEQ_1:28;
    i <= n by A8,A12,FINSEQ_3:25;
    then x+i <= x+n by XREAL_1:6;
    then x+i+1<=x+n+1 by XREAL_1:6;
    then
A14: x+n+1 -' (x+(1+i)) = x+n+1 - (x+(1+i)) by XREAL_1:233;
    1 <= i by A12,FINSEQ_3:25;
    then
A15: n-i <= n-1 by XREAL_1:13;
    thus e.i = c.(1+i) by A7,A9,A12,FINSEQ_1:def 7
      .= r.(x+(1+i)) by A4,A6,A13,FINSEQ_1:def 7
      .= p.(x+(1+i)-'1)*monomial(a,n).(x+n+1-'(x+(1+i))) by A3,A4,A6,A13,
FINSEQ_1:28
      .= p.(x+(1+i)-'1)*0.L by A14,A15,A11,Def5
      .= 0.L;
  end;
  then
A16: Sum e = 0.L by POLYNOM3:1;
  now
    let i be Element of NAT;
A17: dom b c= dom r by A6,FINSEQ_1:26;
    assume
A18: i in dom b;
    then i <= x by A4,FINSEQ_3:25;
    then i+n <= x+n by XREAL_1:6;
    then i+n < x+n+1 by NAT_1:13;
    then
A19: n < x+n+1-i by XREAL_1:20;
    then
A20: x+n+1-i = x+n+1-'i by XREAL_0:def 2;
    thus b.i = r.i by A6,A18,FINSEQ_1:def 7
      .= p.(i-'1)*monomial(a,n).(x+n+1-'i) by A3,A18,A17
      .= p.(i-'1)*0.L by A19,A20,Def5
      .= 0.L;
  end;
  then
A21: Sum b = 0.L by POLYNOM3:1;
A22: 1 in dom d by A7,FINSEQ_3:25;
  then d.1 = c.1 by A9,FINSEQ_1:def 7
    .= r.(x+1) by A4,A6,A22,A10,FINSEQ_1:def 7
    .= p.(x+1-'1)*monomial(a,n).(x+n+1-'(x+1)) by A3,A4,A6,A22,A10,FINSEQ_1:28
    .= p.x*monomial(a,n).(n+(x+1)-'(x+1)) by NAT_D:34
    .= p.x*monomial(a,n).n by NAT_D:34
    .= p.x*a by Def5;
  then d = <* p.x*a *> by A7,FINSEQ_1:40;
  then Sum d = p.x*a by RLVECT_1:44;
  then Sum c = (p.x)*a + 0.L by A9,A16,RLVECT_1:41
    .= (p.x)*a by RLVECT_1:4;
  hence (p*'monomial(a,n)).(x+n) = 0.L + (p.x)*a by A2,A6,A21,RLVECT_1:41
    .= (p.x)*a by RLVECT_1:4;
end;
