reserve a,b,i,k,m,n for Nat;
reserve s,z for non zero Nat;
reserve c for Complex;

theorem
  not 8 divides n
  iff
  ex k st n = 8*k+1 or n = 8*k+2 or n = 8*k+3 or n = 8*k+4 or n = 8*k+5 or
   n = 8*k+6 or n = 8*k+7
  proof
    consider K being Nat such that
A1: n = 8*K or n = 8*K+1 or n = 8*K+2 or n = 8*K+3 or n = 8*K+4 or n = 8*K+5 or
    n = 8*K+6 or n = 8*K+7 by Th28;
    thus not 8 divides n implies
    ex k st n = 8*k+1 or n = 8*k+2 or n = 8*k+3 or n = 8*k+4 or n = 8*k+5 or
     n = 8*k+6 or n = 8*k+7 by A1;
    given k such that
A2: n = 8*k+1 or n = 8*k+2 or n = 8*k+3 or n = 8*k+4 or n = 8*k+5 or n = 8*k+6
    or n = 8*k+7;
    given t being Nat such that
A3: n = 8*t;
    per cases by A2;
    suppose n = 8*k+1;
      then 1/8 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 8*k+2;
      then 2/8 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 8*k+3;
      then 3/8 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 8*k+4;
      then 4/8 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 8*k+5;
      then 5/8 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 8*k+6;
      then 6/8 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 8*k+7;
      then 7/8 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
  end;
