reserve a,b,c for positive Real,
  m,x,y,z for Real,
  n for Nat,
  s,s1,s2,s3,s4,s5 for Real_Sequence;

theorem
  a*b*c = 1 implies (1/a)+(1/b)+(1/c)>=sqrt(a)+sqrt(b)+sqrt(c)
proof
  assume
A1: a*b*c =1;
  (1/a+1/b)/2>=sqrt((1/a)*(1/b)) by Th2;
  then (1/a+1/b)/2>=sqrt(1/(a*b)) by XCMPLX_1:102;
  then
A2: (1/a+1/b)/2>=sqrt((1*c)/((a*b)*c)) by XCMPLX_1:91;
  (1/c+1/a)/2>=sqrt((1/c)*(1/a)) by Th2;
  then (1/c+1/a)/2>=sqrt(1/(c*a)) by XCMPLX_1:102;
  then
A3: (1/c+1/a)/2>=sqrt((1*b)/((c*a)*b)) by XCMPLX_1:91;
  (1/b+1/c)/2>=sqrt((1/b)*(1/c)) by Th2;
  then (1/b+1/c)/2>=sqrt(1/(b*c)) by XCMPLX_1:102;
  then (1/b+1/c)/2>=sqrt((1*a)/((b*c)*a)) by XCMPLX_1:91;
  then (1/b+1/c)/2+(1/c+1/a)/2>=sqrt(a)+sqrt(b) by A1,A3,XREAL_1:7;
  then (1/b+1/c)/2+(1/c+1/a)/2+(1/a+1/b)/2>=sqrt(a)+sqrt(b)+sqrt(c) by A1,A2,
XREAL_1:7;
  hence thesis;
end;
