reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th82:
  for n be positive Nat holds k*(a|^n+1)|^m mod a = k mod a
  proof
    let n be positive Nat;
    per cases by NAT_1:25;
    suppose
      C1: a > 1;
      (a|^n+1)|^m mod a = ((a|^n+1) mod a)|^m mod a by GR_CY_3:30; then
      k*(a|^n+1)|^m mod a = k*((1*(a|^n+1) mod a)|^m mod a) mod a by RADIX_2:3
      .= k*((1 mod a)|^m mod a) mod a by Th81
      .= k*(1|^m mod a) mod a by C1, NAT_D:14
      .= k*1 mod a by RADIX_2:3;
      hence thesis;
    end;
    suppose
      a = 1; then
      k*(a|^n+1)|^m mod a = 0 & k mod a = 0 by RADIX_2:1;
      hence thesis;
    end;
    suppose
      a = 0;
      hence thesis;
    end;
  end;
