
theorem
  for n,k being Nat st n <= k holds dyadic(n) c= dyadic(k)
proof
  let n,k be Nat;
A1: for s being Nat holds dyadic(n) c= dyadic(n+s)
  proof
    defpred P[Nat] means dyadic(n) c= dyadic(n+$1);
A2: for k being Nat st P[k] holds P[(k+1)]
    proof
      let k be Nat;
A3:   dyadic(n+k) c= dyadic(n+k+1) by URYSOHN1:5;
      assume dyadic(n) c= dyadic(n+k);
      hence thesis by A3,XBOOLE_1:1;
    end;
A4: P[0];
    for k be Nat holds P[k] from NAT_1:sch 2(A4,A2);
    hence thesis;
  end;
  assume n <= k;
  then consider s being Nat such that
A5: k = n + s by NAT_1:10;
  thus thesis by A1,A5;
end;
