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
  x>y & y>z implies x^2*y+y^2*z+z^2*x>x*y^2+y*z^2+z*x^2
proof
  assume that
A1: x>y and
A2: y>z;
A3: y-z>0 by A2,XREAL_1:50;
  x>z by A1,A2,XXREAL_0:2;
  then
A4: x-z>0 by XREAL_1:50;
  x-y>0 by A1,XREAL_1:50;
  then (x-y)*(y-z)*(x-z)>0 by A3,A4;
  then (x^2*y+y^2*z+z^2*x)-(x*y^2+y*z^2+z*x^2)>0;
  hence thesis by XREAL_1:47;
end;
