reserve a,b,c,d for positive Real,
  m,u,w,x,y,z for Real,
  n,k for Nat,
  s,s1 for Real_Sequence;

theorem
  (a*b+c*d)*(a*c+b*d)>=4*a*b*c*d
proof
  sqrt(a*d)>0 by SQUARE_1:25;
  then (a+d)^2>=(2*sqrt(a*d))^2 by SIN_COS2:1,SQUARE_1:15;
  then (a+d)^2>=2^2*(sqrt(a*d))^2;
  then (a+d)^2>=2*2*(a*d) by SQUARE_1:def 2;
  then
A1: (a+d)^2*(b*c)>=4*a*d*(b*c) by XREAL_1:64;
  (a*b+c*d)*(a*c+b*d) =((sqrt(a*b))^2+c*d)*(a*c+b*d) by SQUARE_1:def 2
    .=((sqrt(a*b))^2+(sqrt(c*d))^2)*(a*c+b*d) by SQUARE_1:def 2
    .=((sqrt(a*b))^2+(sqrt(c*d))^2)*((sqrt(a*c))^2+b*d) by SQUARE_1:def 2
    .=((sqrt(a*b))^2+(sqrt(c*d))^2)*((sqrt(a*c))^2+(sqrt(b*d))^2) by
SQUARE_1:def 2;
  then (a*b+c*d)*(a*c+b*d)>=b*c*(a+d)^2 by Lm5;
  hence thesis by A1,XXREAL_0:2;
end;
