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

theorem Th34:
  not 6 divides n
  iff
  ex k st n = 6*k+1 or n = 6*k+2 or n = 6*k+3 or n = 6*k+4 or n = 6*k+5
  proof
    consider K being Nat such that
A1: n = 6*K or n = 6*K+1 or n = 6*K+2 or n = 6*K+3 or n = 6*K+4 or n = 6*K+5
    by Th26;
    thus not 6 divides n implies
    ex k st n = 6*k+1 or n = 6*k+2 or n = 6*k+3 or n = 6*k+4 or n = 6*k+5
    by A1;
    given k such that
A2: n = 6*k+1 or n = 6*k+2 or n = 6*k+3 or n = 6*k+4 or n = 6*k+5;
    given t being Nat such that
A3: n = 6*t;
    per cases by A2;
    suppose n = 6*k+1;
      then 1/6 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 6*k+2;
      then 2/6 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 6*k+3;
      then 3/6 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 6*k+4;
      then 4/6 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 6*k+5;
      then 5/6 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
  end;
