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)|^3+(b/c)|^3+(c/a)|^3>=b/a+c/b+a/c
proof
A1: 1=(a/a)*1*1 by XCMPLX_1:60
    .=(a/a)*(b/b)*1 by XCMPLX_1:60
    .=(a/a)*(b/b)*(c/c) by XCMPLX_1:60
    .=(a/b)*(b/a)*(c/c)
    .=(a/b)*((b/a)*(c/c))
    .=(a/b)*((b/c)*(c/a))
    .=(a/b)*(b/c)*(c/a);
  (c/b)=(c/b)*1 .=(c/b)*(a/a) by XCMPLX_1:60
    .=(a/b)*(c/a)*1;
  then
A2: (c/b)<=((a/b)|^3+(c/a)|^3+1|^3)/3 by Th13;
  (a/c)=(a/c)*1 .=(a/c)*(b/b) by XCMPLX_1:60
    .=(a/b)*(b/c)*1;
  then
A3: (a/c)<=((a/b)|^3+(b/c)|^3+1|^3)/3 by Th13;
  (b/a)=(b/a)*1 .=(b/a)*(c/c) by XCMPLX_1:60
    .=(b/c)*(c/a)*1;
  then (b/a)<=((b/c)|^3+(c/a)|^3+1|^3)/3 by Th13;
  then
  (b/a)+(c/b)<=(((b/c)|^3)/3+((c/a)|^3)/3+1/3)+(((a/b)|^3)/3+ ((c/a)|^3)/3
  +1/3) by A2,XREAL_1:7;
  then
A4: (b/a)+(c/b)+(a/c)<=(b/c)|^3/3+(a/b)|^3/3+2*(c/a)|^3/3+2/3+ (((a/b)|^3)/
  3+((b/c)|^3)/3+1/3) by A3,XREAL_1:7;
  (a/b)*(b/c)*(c/a)<=((a/b)|^3+(b/c)|^3+(c/a)|^3)/3 by Th13;
  then (b/a)+(c/b)+(a/c)+1<=2*(b/c)|^3/3+2*(a/b)|^3/3+2*(c/a)|^3/3+1+ (((a/b)
  |^3)/3+((b/c)|^3)/3+((c/a)|^3)/3) by A1,A4,XREAL_1:7;
  then (b/a)+(c/b)+(a/c)+1-1<=(b/c)|^3+(a/b)|^3+(c/a)|^3+1-1 by XREAL_1:9;
  hence thesis;
end;
