reserve a,b,d,n,k,i,j,x,s for Nat;

theorem :: Problem 103
  (for n being Nat st n > 5 ex p,q being Prime st n < p < q < 2*n)
implies
  (for n being Nat st n > 10 ex p,q being Prime st
     p < q & p |-count (n!) = 1 & q |-count (n!) = 1)
proof
  assume
A1: for n being Nat st n > 5 ex p,q being Prime st n < p < q < 2*n;
  let n be Nat such that
A2: n > 10;
  n >= 10+1 by A2,NAT_1:13;
  then per cases by XXREAL_0:1;
  suppose
A3:  n = 11;
    reconsider p=7,q=11 as Prime by XPRIMES1:7,11;
    take p,q;
    1 div 2 = 0 by NAT_D:27;
    then
A4:   5*2+1 div 2 = 5 + 0 by NAT_D:61;
A5:   q|^1 divides q! by NEWTON:41;
    2*11 > 11;
    then not q|^(1+1) divides q! by Th36;
    hence thesis by A4,Th38,A3,A5,NAT_3:def 7;
  end;
  suppose n > 11;
    then
A6:   n >= 11+1 by NAT_1:13;
    set k = n div 2;
    6*2 div 2 = 6 by NAT_D:18;
    then 5+1 <= k by A6,NAT_2:24;
    then 5 < k by NAT_1:13;
    then consider p,q be Prime such that
A7:   k < p < q < 2*k by A1;
    take p,q;
    p <= 2*k & k < q & 2 < n by A2,A7,XXREAL_0:2;
    hence thesis by A7,Th38;
  end;
end;
