reserve p,q for Rational;
reserve g,m,m1,m2,n,n1,n2 for Nat;
reserve i,i1,i2,j,j1,j2 for Integer;
reserve R for Ring, F for Field;

theorem
for p being Prime,
    R being p-characteristic Ring,
    n being positive Nat
holds n is Element of CharSet R iff p divides n
proof
let p be Prime, R be p-characteristic Ring, j be positive Nat;
A1: Char R = p by Def6; then
A2: p '*' 1.R = 0.R by Def5;
A3: now assume A4: j is Element of CharSet R;
   A5: ((j div p) * p) '*' 1.R = ((j div p) '*' 1.R) * (p '*' 1.R) by Th66
                             .= 0.R by A2;
   A6: j '*' 1.R = ((j div p) * p + (j mod p)) '*' 1.R by INT_1:59
               .= 0.R + ((j mod p) '*' 1.R) by A5,Th61
               .= (j mod p) '*' 1.R;
   j in {k where k is positive Nat : k '*' 1.R = 0.R} by A4;
   then consider l being positive Nat such that A7: l = j & l '*' 1.R = 0.R;
   now assume j mod p > 0;
     then reconsider l = j mod p as positive Nat;
     A8: l in {k where k is positive Nat : k '*' 1.R = 0.R} by A7,A6;
     p = min(CharSet R) by A1,Th78;
     then p <= (j mod p) by A8,XXREAL_2:def 7;
     hence contradiction by INT_1:58;
     end;
   then A9: j mod p = 0;
   j  = (j div p) * p + (j mod p) by INT_1:59;
   hence p divides j by A9;
   end;
now assume p divides j;
  then consider i being Integer such that A10: j = p * i;
  j '*' 1.R = (p '*' 1.R) * (i '*' 1.R) by A10,Th66
           .= 0.R * (i '*' 1.R) by A1,Def5
           .= 0.R;
  then j in {k where k is positive Nat : k '*' 1.R = 0.R};
  hence j is Element of CharSet R;
  end;
hence thesis by A3;
end;
