theorem Th80:
  f|Y is bounded implies (r(#)f)|Y is bounded
proof
  assume
A1: f|Y is bounded;
  per cases;
  suppose
A2: 0 <= r;
    hence (r(#)f)|Y is bounded_above by A1,Th78;
    thus thesis by A1,A2,Th79;
  end;
  suppose
A3: r <= 0;
    hence (r(#)f)|Y is bounded_above by A1,Th79;
    thus thesis by A1,A3,Th78;
  end;
end;
