reserve a,b,c,d,x,j,k,l,m,n for Nat,
  p,q,t,z,u,v for Integer,
  a1,b1,c1,d1 for Complex;

theorem
  a|^(m+1)+b|^(m+1) >= (a|^m+b|^m)*(a+b)/2
  proof
    A1: a|^(m+1)+b|^(m+1)=((a|^m+b|^m)*(a+b) + (a-b)*(a|^m-b|^m))/2 by Th9
    .= (a|^m+b|^m)*(a+b)/2 + (a-b)*(a|^m-b|^m)/2;
    (a|^m+b|^m)*(a+b) + (a-b)*(a|^m-b|^m) >= (a|^m+b|^m)*(a+b)
    by Th10,XREAL_1:31; then
    ((a|^m+b|^m)*(a+b) + (a-b)*(a|^m-b|^m))/2 >= ((a|^m+b|^m)*(a+b))/2
    by XREAL_1:72;
    hence thesis by A1;
  end;
