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
  a>b & b>c implies (a to_power (2*a))*(b to_power (2*b))*(c to_power (2
  *c)) > (a to_power (b+c))*(b to_power (a+c))*(c to_power (a+b))
proof
  assume that
A1: a>b and
A2: b>c;
A3: b/c>1 by A2,XREAL_1:187;
  set e=(a to_power (b+c))*(b to_power (a+c))*(c to_power (a+b));
A4: b to_power (c+a)>0 by POWER:34;
  b-c>0 by A2,XREAL_1:50;
  then
A5: (b/c) to_power (b-c) > 1 by A3,POWER:35;
A6: a/b>1 by A1,XREAL_1:187;
  a-b>0 by A1,XREAL_1:50;
  then (a/b) to_power (a-b) > 1 by A6,POWER:35;
  then
A7: ((a/b) to_power (a-b))*((b/c) to_power (b-c)) > 1 by A5,XREAL_1:161;
A8: c to_power (a+b)>0 by POWER:34;
A9: a>c by A1,A2,XXREAL_0:2;
  then
A10: a/c>1 by XREAL_1:187;
  set d=(a to_power (2*a))*(b to_power (2*b))*(c to_power (2*c));
A11: a to_power (b+c)>0 by POWER:34;
  a-c>0 by A9,XREAL_1:50;
  then (a/c) to_power (a-c) > 1 by A10,POWER:35;
  then ((a/b) to_power (a-b))*((b/c) to_power (b-c)) *((a/c) to_power -(c-a))
  >1 by A7,XREAL_1:161;
  then
  ((a/b) to_power (a-b))*((b/c) to_power (b-c)) *((c/a) to_power (c-a)) >
  1 by Lm4;
  then (a to_power (a-b)/b to_power (a-b))*((b/c) to_power (b-c)) *((c/a)
  to_power (c-a)) >1 by POWER:31;
  then (a to_power (a-b)/b to_power (a-b))*(b to_power (b-c)/c to_power (b-c)
  ) *((c/a) to_power (c-a)) >1 by POWER:31;
  then (a to_power (a-b)/b to_power (a-b))*(b to_power (b-c)/c to_power (b-c)
  ) *(c to_power (c-a)/a to_power (c-a)) >1 by POWER:31;
  then
  ((a to_power (a-b))*(b to_power (b-c)))/((c to_power (b-c))*(b to_power
  ( a-b))) *(c to_power (c-a)/a to_power (c-a)) >1 by XCMPLX_1:76;
  then
  ((a to_power (a-b))/(c to_power (b-c)))*((b to_power (b-c))/(b to_power
  ( a-b))) *(c to_power (c-a)/a to_power (c-a)) >1 by XCMPLX_1:76;
  then ((a to_power (a-b))/(c to_power (b-c)))*(b to_power ((b-c)-(a-b))) *(c
  to_power (c-a)/a to_power (c-a)) >1 by POWER:29;
  then
  ((a to_power (a-b))/(c to_power (b-c)))*(c to_power (c-a)/a to_power (c
  -a)) *(b to_power (2*b-a-c)) >1;
  then
  ((a to_power (a-b))/a to_power (c-a))*(c to_power (c-a)/(c to_power ( b
  -c))) *(b to_power (2*b-a-c)) >1 by XCMPLX_1:85;
  then (a to_power ((a-b)-(c-a))) *(c to_power (c-a)/(c to_power (b-c))) *(b
  to_power (2*b-a-c)) >1 by POWER:29;
  then
  (a to_power (2*a-b-c)) *(c to_power ((c-a)-(b-c))) *(b to_power (2*b-a-
  c)) >1 by POWER:29;
  then
  (a to_power (2*a-(b+c))) *(c to_power (2*c-(a+b))) *(b to_power (2*b-(a
  +c))) >1;
  then (a to_power (2*a)/a to_power (b+c))*(c to_power (2*c-(a+b))) *(b
  to_power (2*b-(a+c))) >1 by POWER:29;
  then
  (a to_power (2*a)/a to_power (b+c))*(c to_power (2*c)/c to_power (a+b )
  ) *(b to_power (2*b-(a+c))) >1 by POWER:29;
  then ((a to_power (2*a))*((c to_power (2*c)))/((a to_power (b+c))* (c
  to_power (a+b))))*(b to_power (2*b-(a+c))) >1 by XCMPLX_1:76;
  then ((a to_power (2*a))*((c to_power (2*c)))/((a to_power (b+c))* (c
  to_power (a+b))))* (b to_power (2*b)/b to_power (a+c)) >1 by POWER:29;
  then ((a to_power (2*a))*(c to_power (2*c))*(b to_power (2*b)))/ ((a
  to_power (b+c))* (c to_power (a+b))*(b to_power (a+c))) >1 by XCMPLX_1:76;
  then (d/e)*e>1*e by A11,A4,A8,XREAL_1:68;
  hence thesis by A11,A4,A8,XCMPLX_1:87;
end;
