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 Th83:
  for n be positive Nat holds
    b*(a|^n+1)|^m + c*(a|^n+1)|^l mod a = b+c mod a
  proof
    let n be positive Nat;
    A1: b*(a|^n+1)|^m mod a = b mod a & c*(a|^n+1)|^l mod a = c mod a by Th82;
    b*(a|^n+1)|^m + c*(a|^n+1)|^l mod a
    = ((b mod a) + (c mod a))mod a by A1,NAT_D:66
    .= b+c mod a by NAT_D:66;
    hence thesis;
  end;
