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

theorem Th32:
  not 4 divides n iff ex k st n = 4*k+1 or n = 4*k+2 or n = 4*k+3
  proof
    consider K being Nat such that
A1: n = 4*K or n = 4*K+1 or n = 4*K+2 or n = 4*K+3 by Th24;
    thus not 4 divides n implies ex k st n = 4*k+1 or n = 4*k+2 or n = 4*k+3
    by A1;
    given k such that
A2: n = 4*k+1 or n = 4*k+2 or n = 4*k+3;
    given t being Nat such that
A3: n = 4*t;
    per cases by A2;
    suppose n = 4*k+1;
      then 1/4 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 4*k+2;
      then 2/4 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 4*k+3;
      then 3/4 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
  end;
