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

theorem Th22:
  ex k st n = z*k+(0 qua Nat) or ... or n = z*k+(z-1)
  proof
    per cases;
    suppose
A1:   z-1 <= n;
      n,0 are_congruent_mod z or ... or n,z-1 are_congruent_mod z by NAT_6:13;
      then consider i being Nat such that
      0 <= i and
A2:   i <= z-1 and
A3:   n,i are_congruent_mod z;
      consider k being Integer such that
A4:   z*k = n-i by A3;
      n-(z-1) >= n-n by A1,XREAL_1:10;
      then z*k >= 0 by A2,A4,XREAL_1:10;
      then k is non negative;
      then reconsider k as Nat by TARSKI:1;
      take k;
      thus n = z*k+(0 qua Nat) or ... or n = z*k+(z-1)
      proof
        take i;
        thus thesis by A2,A4;
      end;
    end;
    suppose
A5:   n < z-1;
      take k = 0;
      thus n = z*k+(0 qua Nat) or ... or n = z*k+(z-1)
      proof
        take n;
        thus thesis by A5;
      end;
    end;
  end;
