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+1/a)*(b+1/b)>=(sqrt(a*b)+1/sqrt(a*b))^2
proof
A1: sqrt(a*b)>0 by SQUARE_1:25;
A2: (a+1/a)*(b+1/b)=a*b+a*(1/b)+(1/a)*b+(1/a)*(1/b)
    .=a*b+(a*1)/b+(1/a)*b+(1/a)*(1/b) by XCMPLX_1:74
    .=a*b+(a*1)/b+(b*1)/a+(1/a)*(1/b) by XCMPLX_1:74
    .=a*b+(a*1)/b+(b*1)/a+(1*1)/(a*b) by XCMPLX_1:76
    .=a*b+a/b+b/a+1/(a*b);
  a/b+b/a>=2*sqrt((a/b)*(b/a)) by SIN_COS2:1;
  then a/b+b/a>=2*sqrt((a*b)/(b*a)) by XCMPLX_1:76;
  then a/b+b/a>=2*1 by SQUARE_1:18,XCMPLX_1:60;
  then
A3: (a/b+b/a)+(a*b+1/(a*b))>=2+(a*b+1/(a*b)) by XREAL_1:6;
  (sqrt(a*b)+1/sqrt(a*b))^2 =(sqrt(a*b))^2+2*sqrt(a*b)*(1/sqrt(a*b))+(1/
  sqrt(a*b))^2
    .=a*b+2*sqrt(a*b)*(1/sqrt(a*b))+(1/sqrt(a*b))^2 by SQUARE_1:def 2
    .=a*b+2*sqrt(a*b)*(1/sqrt(a*b))+1^2/(sqrt(a*b))^2 by XCMPLX_1:76
    .=a*b+2*(sqrt(a*b)*(1/sqrt(a*b)))+1/(a*b) by SQUARE_1:def 2
    .=a*b+2*((sqrt(a*b)*1)/sqrt(a*b))+1/(a*b) by XCMPLX_1:74
    .=a*b+2*(1)+1/(a*b) by A1,XCMPLX_1:60
    .=a*b+2+1/(a*b);
  hence thesis by A2,A3;
end;
