reserve a,b,k,m,n,s for Nat;
reserve c,c1,c2,c3 for Complex;
reserve i,j,z for Integer;
reserve p for Prime;
reserve x for object;

theorem Th20:
  n > 1 implies
   ex p be Prime st p divides n &
     for q be Prime st q divides n holds q <= p
  proof
    assume
A1: n > 1;
    defpred P[Nat] means $1 is prime & $1 divides n;
A2: for k being Nat st P[k] holds k <= n by A1,NAT_D:7;
    n>=1+1 by A1,NAT_1:13;
    then ex p being Element of NAT st p is prime & p divides n by INT_2:31;
    then
A3: ex k being Nat st P[k];
    consider k be Nat such that
A4: P[k] & for n being Nat st P[n]
    holds n <= k from NAT_1:sch 6(A2,A3);
    reconsider k as Prime by A4;
    take k;
    thus thesis by A4;
  end;
