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 & b>=c implies (a to_power a)*(b to_power b)*(c to_power c)>=(a*b
  *c) to_power ((a+b+c)/3)
proof
  assume that
A1: a>=b and
A2: b>=c;
A3: ((b/c) to_power ((b-c)/3)) =((b/c) #R ((b-c)/3)) by POWER:def 2;
  b/c>=1 & b-c>=c-c by A2,XREAL_1:9,181;
  then
A4: ((b/c) to_power ((b-c)/3))>=1 by A3,PREPOWER:85;
A5: ( (a/b) to_power ((a-b)/3))=((a/b) #R ((a-b)/3)) by POWER:def 2;
  a/b>=1 & a-b>=b-b by A1,XREAL_1:9,181;
  then ((a/b) to_power ((a-b)/3))>=1 by A5,PREPOWER:85;
  then
A6: ((a/b) to_power ((a-b)/3))*((b/c) to_power ((b-c)/3))>=1*1 by A4,XREAL_1:66
;
  a>=c by A1,A2,XXREAL_0:2;
  then
A7: a/c>=1 & a-c>=c-c by XREAL_1:9,181;
  ((a/c) to_power ((a-c)/3))=((a/c) #R ((a-c)/3)) by POWER:def 2;
  then ((a/c) to_power ((a-c)/3))>=1 by A7,PREPOWER:85;
  then ((a/b) to_power ((a-b)/3))*((b/c) to_power ((b-c)/3))* ((a/c) to_power
  ((a-c)/3))>=1 by A6,XREAL_1:66;
  then ((a to_power ((a-b)/3))/(b to_power ((a-b)/3)))*((b/c) to_power ((b-c)
  /3))* ((a/c) to_power ((a-c)/3))>=1 by POWER:31;
  then
A8: ((a to_power ((a-b)/3))/(b to_power ((a-b)/3)))*((b to_power ((b-c)/3)
  )/ (c to_power ((b-c)/3)))*((a/c) to_power ((a-c)/3))>=1 by POWER:31;
  set t=b to_power ((a+b+c)/3);
  set s=b to_power b;
  set r=c to_power ((a+b+c)/3);
  set q=c to_power c;
  set p=a to_power ((a+b+c)/3);
  set o=a to_power a;
  set j=c to_power ((a-c)/3);
  set i=a to_power ((a-c)/3);
  set h=c to_power ((b-c)/3);
  set w=b to_power ((b-c)/3);
  set v=b to_power ((a-b)/3);
  set u=a to_power ((a-b)/3);
A9: p>0 & r>0 by POWER:34;
A10: t>0 by POWER:34;
  ((a to_power ((a-b)/3))/(b to_power ((a-b)/3)))*((b to_power ((b-c)/3))
/ (c to_power ((b-c)/3)))*((a to_power ((a-c)/3))/(c to_power ((a-c)/3))) =(u*w
  )/(v*h)*(i/j) by XCMPLX_1:76
    .=(u*w*i)/(v*h*j) by XCMPLX_1:76
    .=((a to_power ((a-b)/3))*(a to_power ((a-c)/3))*(b to_power ((b-c)/3)))
  / ((b to_power ((a-b)/3))*(c to_power ((a-c)/3))*(c to_power ((b-c)/3)))
    .=((a to_power (((a-b)/3)+((a-c)/3)))*(b to_power ((b-c)/3)))/ ((b
to_power ((a-b)/3))*(c to_power ((a-c)/3))*(c to_power ((b-c)/3))) by POWER:27
    .=((a to_power ((2*a-b-c)/3))*(b to_power ((b-c)/3)))/ ((b to_power ((a-
  b)/3))*((c to_power ((a-c)/3))*(c to_power ((b-c)/3))))
    .=((a to_power ((2*a-b-c)/3))*(b to_power ((b-c)/3)))/ ((b to_power ((a-
  b)/3))*((c to_power (((a-c)/3)+((b-c)/3))))) by POWER:27
    .=((a to_power ((2*a-b-c)/3))/(c to_power ((a+b-2*c)/3)))* ((b to_power
  ((b-c)/3))/(b to_power ((a-b)/3))) by XCMPLX_1:76
    .=((a to_power ((2*a-b-c)/3))/(c to_power ((a+b-2*c)/3)))* (b to_power (
  (b-c)/3-(a-b)/3)) by POWER:29
    .=(1/(c to_power ((a+b-2*c)/3)))* (b to_power ((2*b-a-c)/3))*(a to_power
  ((2*a-b-c)/3))
    .=(c to_power (-((a+b-2*c)/3)))* (b to_power ((2*b-a-c)/3))*(a to_power
  ((2*a-b-c)/3)) by POWER:28
    .=(a to_power ((3*a)/3-(a+b+c)/3))*(c to_power ((3*c)/3-(a+b+c)/3))* (b
  to_power ((3*b)/3-(a+b+c)/3))
    .=((a to_power ((3*a)/3))/(a to_power ((a+b+c)/3))) *(c to_power ((3*c)/
  3-(a+b+c)/3))* (b to_power ((3*b)/3-(a+b+c)/3)) by POWER:29
    .=((a to_power ((3*a)/3))/(a to_power ((a+b+c)/3))) *((c to_power ((3*c)
  /3))/(c to_power ((a+b+c)/3)))* (b to_power ((3*b)/3-(a+b+c)/3)) by POWER:29
    .=((a to_power a)/(a to_power ((a+b+c)/3)))* ((c to_power c)/(c to_power
  ((a+b+c)/3)))* ((b to_power b)/(b to_power ((a+b+c)/3))) by POWER:29
    .=(o*q)/(p*r)*(s/t) by XCMPLX_1:76
    .=((a to_power a)*(c to_power c)*(b to_power b))/((a to_power ((a+b+c)/3
  )) *(c to_power ((a+b+c)/3))*(b to_power ((a+b+c)/3))) by XCMPLX_1:76;
  then (o*q*s)/(p*r*t)>=1 by A8,POWER:31;
  then (o*q*s)/(p*r*t)*(p*r*t)>=1*(p*r*t) by A9,A10,XREAL_1:64;
  then ((a to_power a)*(c to_power c)*(b to_power b))>=((a to_power ((a+b+c)/
  3)) *(c to_power ((a+b+c)/3))*(b to_power ((a+b+c)/3))) by A9,A10,XCMPLX_1:87
;
  then
  (a to_power a)*(c to_power c)*(b to_power b)>= ((a*c) to_power ((a+b+c)
  /3))*(b to_power ((a+b+c)/3)) by POWER:30;
  then
  (a to_power a)*((b to_power b)*(c to_power c))>=(a*c*b) to_power ((a+b+
  c)/3) by POWER:30;
  hence thesis;
end;
