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 + b|^m <= c|^m implies ex x st a|^m + b|^m <= (a+x)|^m
  proof
    assume
    A1: a|^m +b|^m <= c|^m;
    then ex x st c = a + x by Th6,NAT_1:10;
    hence thesis by A1;
  end;
