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)*(1/a+1/b)>=4
proof
  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*sqrt((a/b*b)/a) by XCMPLX_1:83;
  then a/b+b/a>=2*sqrt(a/a) by XCMPLX_1:87;
  then a/b+b/a>=2*1 by SQUARE_1:18,XCMPLX_1:60;
  then (a/b+b/a)+2>=2+2 by XREAL_1:7;
  then a/b+b/a+1+1>=4;
  then a*(1/b)+b/a+1+1>=4 by XCMPLX_1:99;
  then a*(1/b)+b*(1/a)+1+1>=4 by XCMPLX_1:99;
  then a*(1/b)+b*(1/a)+a*(1/a)+1>=4 by XCMPLX_1:106;
  then a*(1/b)+b*(1/a)+a*(1/a)+b*(1/b)>=4 by XCMPLX_1:106;
  hence thesis;
end;
