reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  n > 4 implies
  ex m being Nat st n < m < 2*n & m is_a_product_of_two_different_primes
  proof
    assume n > 4;
    then consider k being Nat such that
A1: n = 2*k & k > 2 or n = 2*k+1 & k > 1 by Th37;
    reconsider k as Element of NAT by ORDINAL1:def 12;
    reconsider p2 = 2 as Prime by XPRIMES1:2;
    per cases by A1;
    suppose that
A2:   n = 2*k and
A3:   k > 2;
      consider P being Prime such that
A4:   k < P and
A5:   P <= 2*k by A3,XXREAL_0:2,NAT_4:56;
      take m = p2*P;
      thus n < m by A2,A4,XREAL_1:68;
      2*k is composite by A3,Th3;
      then P < n by A2,A5,XXREAL_0:1;
      hence m < 2*n by XREAL_1:68;
      take p2,P;
      thus p2 <> P by A3,A4;
      thus m = p2*P;
    end;
    suppose that
A6:   n = 2*k+1 and
A7:   k > 1;
      consider P being Prime such that
A8:   k < P and
A9:   P <= 2*k by A7,NAT_4:56;
      take m = p2*P;
      k+1 <= P by A8,NAT_1:13;
      then
A10:  2*(k+1) <= 2*P by XREAL_1:64;
      2*k+1 < 2*k+2 by XREAL_1:6;
      hence n < m by A6,A10,XXREAL_0:2;
A11:  2*P <= 2*(2*k) by A9,XREAL_1:64;
      4*k+0 < 4*k+2 by XREAL_1:6;
      hence m < 2*n by A6,A11,XXREAL_0:2;
      take p2,P;
      k >= 1+1 by A7,NAT_1:13;
      hence p2 <> P by A8;
      thus m = p2*P;
      thus thesis;
    end;
  end;
