theorem
  p <= q implies SetPrimenumber p c= SetPrimenumber q
proof
  assume
A1: p <= q;
  let x be object;
  assume
A2: x in SetPrimenumber p;
  then reconsider x9 = x as Element of NAT;
  x9 < p by A2,Def7;
  then
A3: x9 < q by A1,XXREAL_0:2;
  x9 is prime by A2,Def7;
  hence thesis by A3,Def7;
end;
