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 Th18:
  c|^m >= a|^m + b|^m & a>0 & b>0 & k>0 implies
  c|^(k+m) > a|^(k+m) + b|^(k+m)
  proof
    assume
    A0: c|^m >= a|^m + b|^m & a>0 & b>0 & k>0;
    then consider l such that
    A0b: k = 1 + l by NAT_1:10,14;
    defpred P[Nat] means c|^($1+m+1) >a|^($1+m+1)+b|^($1+m+1) & a>0 & b>0;
    A1: P[0] by A0,Th17;
    A2: P[x] implies P[x+1] by Th17;
    for j holds P[j] from NAT_1:sch 2(A1,A2);
    then c|^(l+m+1) > a|^(l+m+1) + b|^(l+m+1);
    hence thesis by A0b;
  end;
