reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th40:
  for c st c > 0
  for r,s be non negative Real holds r < s iff r|^c < s|^c
  proof
    let c such that
    A0: c > 0;
    let r,s be non negative Real;
    (r < s implies r|^c < s|^c) & (r|^c < s|^c implies r < s)
    proof
      A0a: r < s implies r|^c < s|^c
      proof
        assume
        A1: r < s; then
        s|^c > 0 by PREPOWER:6; then
        A2: r = 0 implies s|^c > r|^c by A0,NAT_1:14,NEWTON:11;
        r>0 implies r to_power c < s to_power c by A0,A1,POWER:37;
        hence thesis by A2;
      end;
      s = 0 implies s|^c = 0 by A0,NAT_1:14,NEWTON:11; then
      r > 0 & s = 0 implies r|^c > s|^c by PREPOWER:6;
      hence thesis by A0a,PREPOWER:9;
    end;
    hence thesis;
  end;
