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 Th7:
  2/((1/a)+(1/b))<=sqrt(a*b)
proof
A1: sqrt(a*b)>0 by SQUARE_1:25;
  then (2*sqrt(a*b))/(a+b)<=1 by SIN_COS2:1,XREAL_1:183;
  then ((2*sqrt(a*b))/(a+b))*sqrt(a*b)<=1*sqrt(a*b) by A1,XREAL_1:64;
  then (2*sqrt(a*b))/((a+b)/sqrt(a*b))<=sqrt(a*b) by XCMPLX_1:82;
  then ((2*sqrt(a*b))*sqrt(a*b))/(a+b)<=sqrt(a*b) by XCMPLX_1:77;
  then (2*(sqrt(a*b))^2)/(a+b)<=sqrt(a*b);
  then 2*(a*b)/(a+b)<=sqrt(a*b) by SQUARE_1:def 2;
  hence thesis by Lm2;
end;
