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 Th15:
  (a+b+c)>=3*(3-root(a*b*c))
proof
A1: (3 -Root(c))>0 by PREPOWER:def 2;
  (3 -Root(a))>0 & (3 -Root(b))>0 by PREPOWER:def 2;
  then (3 -Root(a))|^3+(3 -Root(b))|^3+(3 -Root(c))|^3>=3*(3 -Root(a)) *(3
  -Root(b))*(3 -Root(c)) by A1,Th12;
  then
  a+(3 -Root(b))|^3+(3 -Root(c))|^3>=3*(3 -Root(a)) *(3 -Root(b))*(3 -Root
  (c)) by PREPOWER:19;
  then a+b+(3 -Root(c))|^3>=3*(3 -Root(a))*(3 -Root(b))*(3 -Root(c)) by
PREPOWER:19;
  then a+b+c>=3*((3 -Root(a))*(3 -Root(b)))*(3 -Root(c)) by PREPOWER:19;
  then a+b+c>=3*(3 -Root(a*b))*(3 -Root(c)) by PREPOWER:22;
  then a+b+c>=3*((3 -Root(a*b))*(3 -Root(c)));
  then a+b+c>=3*(3 -Root(a*b*c)) by PREPOWER:22;
  hence thesis by POWER:def 1;
end;
