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 Th20:
  a/b+b/c+c/a>=3
proof
  a/b+b/c+c/a>=3*(3-root((a/b)*(b/c)*(c/a))) by SERIES_3:15;
  then a/b+b/c+c/a>=3*(3-root(((a*b)/(b*c))*(c/a))) by XCMPLX_1:76;
  then a/b+b/c+c/a>=3*(3-root((a/c)*(c/a))) by XCMPLX_1:91;
  then a/b+b/c+c/a>=3*(3-root(((a*c)/(c*a)))) by XCMPLX_1:76;
  then a/b+b/c+c/a>=3*(3-root(1)) by XCMPLX_1:60;
  then a/b+b/c+c/a>=3*1 by POWER:6;
  hence thesis;
end;
