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
  sqrt((a+b)*(c+d))>=sqrt(a*c)+sqrt(b*d)
proof
A1: sqrt((a+b)*(c+d))>0 by SQUARE_1:25;
A2: sqrt(b*d)>0 by SQUARE_1:25;
  a*d+b*c>=2*sqrt((a*d)*(b*c)) by SIN_COS2:1;
  then (a*d+b*c)+(a*c+b*d)>=2*sqrt((a*d)*(b*c))+(a*c+b*d) by XREAL_1:7;
  then (a+b)*(c+d)>=2*sqrt((a*d)*(b*c))+a*c+b*d;
  then (a+b)*(c+d)>=2*sqrt((a*d)*(b*c))+(sqrt(a*c))^2+b*d by SQUARE_1:def 2;
  then (a+b)*(c+d)>=2*sqrt((a*d)*(b*c))+(sqrt(a*c))^2+(sqrt(b*d))^2 by
SQUARE_1:def 2;
  then (sqrt((a+b)*(c+d)))^2>=2*sqrt((a*c)*(b*d))+(sqrt(a*c))^2+(sqrt(b*d))^2
  by SQUARE_1:def 2;
  then
A3: (sqrt((a+b)*(c+d)))^2>=(sqrt(a*c))^2+2*(sqrt(a*c)*sqrt(b*d))+(sqrt(b *d)
  )^2 by SQUARE_1:29;
  (sqrt(a*c)+sqrt(b*d))^2>=0 by XREAL_1:63;
  then
A4: sqrt((sqrt((a+b)*(c+d)))^2)>=sqrt((sqrt(a*c)+sqrt(b*d))^2) by A3,
SQUARE_1:26;
  sqrt(a*c)>0 by SQUARE_1:25;
  then sqrt((sqrt((a+b)*(c+d)))^2)>=sqrt(a*c)+sqrt(b*d) by A4,A2,SQUARE_1:22;
  hence thesis by A1,SQUARE_1:22;
end;
