reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve r for Real;
reserve c for Complex;
reserve e1,e2,e3,e4,e5 for ExtReal;

theorem Th12:
  n divides 8 implies n in {1,2,4,8}
  proof
    assume
A1: n divides 8;
    then n <= 8 by INT_2:27;
    then
A2: n = 0 or ... or n = 8;
    n <> 0 by A1;
    then
A3: 8 mod n = 0 by A1,INT_1:62;
    (3*2+2) mod 3 = 2 mod 3 & (5*1+3) mod 5 = 3 mod 5 &
    (6*1+2) mod 6 = 2 mod 6 & (7*1+1) mod 7 = 1 mod 7 by NAT_D:21;
    hence thesis by A1,A2,A3,ENUMSET1:def 2,NAT_D:24;
  end;
