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

theorem
  not 5 divides n
  iff
  ex k st n = 5*k+1 or n = 5*k+2 or n = 5*k+3 or n = 5*k+4
  proof
    consider K being Nat such that
A1: n = 5*K or n = 5*K+1 or n = 5*K+2 or n = 5*K+3 or n = 5*K+4 by Th25;
    thus not 5 divides n implies
    ex k st n = 5*k+1 or n = 5*k+2 or n = 5*k+3 or n = 5*k+4 by A1;
    given k such that
A2: n = 5*k+1 or n = 5*k+2 or n = 5*k+3 or n = 5*k+4;
    given t being Nat such that
A3: n = 5*t;
    per cases by A2;
    suppose n = 5*k+1;
      then 1/5 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 5*k+2;
      then 2/5 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 5*k+3;
      then 3/5 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
    suppose n = 5*k+4;
      then 4/5 = t-k by A3;
      hence thesis by NAT_D:33;
    end;
  end;
