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|^4+b|^4>=a|^3*b+a*b|^3
proof
  (a-b)^2>=0 by XREAL_1:63;
  then ((a-b)*(a-b))*(a^2+a*b+b^2)>=0;
  then (a-b)*((a-b)*(a^2+a*b+b^2))>=0;
  then (a-b)*(a|^3-b|^3)>=0 by Lm1;
  then a|^3*a-a|^3*b-b|^3*(a-b)>=0;
  then a|^(3+1)-a|^3*b-b|^3*(a-b)>=0 by NEWTON:6;
  then a|^4-a|^3*b-(b|^3*a-b|^3*b)>=0;
  then a|^4-a|^3*b-(b|^3*a-b|^(3+1))>=0 by NEWTON:6;
  then (a|^4+b|^4-a|^3*b-b|^3*a)+(a|^3*b+b|^3*a)>=0+(a|^3*b+b|^3*a) by
XREAL_1:7;
  hence thesis;
end;
