reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;

theorem Th9:
  n divides 9 implies n = 1 or n = 3 or n = 9
  proof
    assume
A1: n divides 9;
    then n <= 9 by INT_2:27;
    then
A2: n = 0 or ... or n = 9;
    now
      (2*4+1) mod 2 = 1 mod 2 by NAT_D:21
      .= 1 by NAT_D:24;
      hence not 2 divides 9;
      (4*2+1) mod 4 = 1 mod 4 by NAT_D:21
      .= 1 by NAT_D:24;
      hence not 4 divides 9 by INT_1:62;
      (5*1+4) mod 5 = 4 mod 5 by NAT_D:21
      .= 4 by NAT_D:24;
      hence not 5 divides 9 by INT_1:62;
      (6*1+3) mod 6 = 3 mod 6 by NAT_D:21
      .= 3 by NAT_D:24;
      hence not 6 divides 9 by INT_1:62;
      (7*1+2) mod 7 = 2 mod 7 by NAT_D:21
      .= 2 by NAT_D:24;
      hence not 7 divides 9 by INT_1:62;
      (8*1+1) mod 8 = 1 mod 8 by NAT_D:21
      .= 1 by NAT_D:24;
      hence not 8 divides 9 by INT_1:62;
    end;
    hence thesis by A1,A2;
  end;
