
theorem Th19:
  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 (monomial(a,n)*'p).(x+n) = a * (p.x
  )
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: (monomial(a,n)*'p).(x+n) = Sum r and
A3: for k be Element of NAT st k in dom r holds r.k = monomial(a,n).(k-'
  1) * p.(x+n+1-'k) by POLYNOM3:def 9;
  len r = n+(1+x) by A1;
  then consider b,c being FinSequence of the carrier of L such that
A4: len b = n and
A5: len c = 1+x 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
  len e = x and
A8: c = d^e by A5,FINSEQ_2:23;
A9: dom d c= dom c by A8,FINSEQ_1:26;
  now
A10: dom b c= dom r by A6,FINSEQ_1:26;
    let i be Element of NAT;
A11: i - 1 < i by XREAL_1:146;
    assume
A12: i in dom b;
    then
A13: i <= n by A4,FINSEQ_3:25;
    1 <= i by A12,FINSEQ_3:25;
    then
A14: i-'1 = i-1 by XREAL_1:233;
    thus b.i = r.i by A6,A12,FINSEQ_1:def 7
      .= monomial(a,n).(i-'1)*p.(x+n+1-'i) by A3,A12,A10
      .= 0.L * p.(x+n+1-'i) by A13,A14,A11,Def5
      .= 0.L;
  end;
  then
A15: Sum b = 0.L by POLYNOM3:1;
  now
    let i be Element of NAT;
A16: n+(1+i) -' 1 = n+i+1 -' 1 .= n+i by NAT_D:34;
    assume
A17: i in dom e;
    then
A18: 1+i in dom c by A7,A8,FINSEQ_1:28;
    i >= 1 by A17,FINSEQ_3:25;
    then n+i >= n+1 by XREAL_1:6;
    then
A19: n+i > n by NAT_1:13;
    thus e.i = c.(1+i) by A7,A8,A17,FINSEQ_1:def 7
      .= r.(n+(1+i)) by A4,A6,A18,FINSEQ_1:def 7
      .= monomial(a,n).(n+(1+i)-'1)*p.(x+n+1-'(n+(1+i))) by A3,A4,A6,A18,
FINSEQ_1:28
      .= 0.L * p.(x+n+1-'(n+(1+i))) by A19,A16,Def5
      .= 0.L;
  end;
  then
A20: Sum e = 0.L by POLYNOM3:1;
A21: 1 in dom d by A7,FINSEQ_3:25;
  then d.1 = c.1 by A8,FINSEQ_1:def 7
    .= r.(n+1) by A4,A6,A21,A9,FINSEQ_1:def 7
    .= monomial(a,n).(n+1-'1)*p.(x+n+1-'(n+1)) by A3,A4,A6,A21,A9,FINSEQ_1:28
    .= monomial(a,n).n*p.(x+(n+1)-'(n+1)) by NAT_D:34
    .= monomial(a,n).n*p.x by NAT_D:34
    .= a*p.x by Def5;
  then d = <* a*p.x *> by A7,FINSEQ_1:40;
  then Sum d = a * (p.x) by RLVECT_1:44;
  then Sum c = a * (p.x) + 0.L by A8,A20,RLVECT_1:41
    .= a * (p.x) by RLVECT_1:4;
  hence (monomial(a,n)*'p).(x+n) = 0.L + a * (p.x) by A2,A6,A15,RLVECT_1:41
    .= a * (p.x) by RLVECT_1:4;
end;
