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
  c|^n > a|^n + b|^n implies
  c|^(k+n) > a|^(k+n) + b|^(k+n)
  proof
    assume
    AX: c|^n > a|^n + b|^n;
    then n <> 0 by Lm0;
    then consider m such that
    AY: n = 1 + m by NAT_1:10,14;
    defpred P[Nat] means c|^($1+m+1) >a|^($1+m+1)+b|^($1+m+1);
    A1: P[0] by AX,AY;
    A2: P[x] implies P[x+1] by Lm29;
    for j holds P[j] from NAT_1:sch 2(A1,A2);
    then c|^(k+m+1) > a|^(k+m+1) + b|^(k+m+1);
    hence thesis by AY;
  end;
