reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem
  A1 is monotone implies A1 (\) A is monotone
proof
  assume
A1: A1 is monotone;
  per cases by A1,SETLIM_1:def 1;
  suppose
    A1 is non-ascending;
    then A1 (\) A is non-ascending by Th30;
    hence thesis by SETLIM_1:def 1;
  end;
  suppose
    A1 is non-descending;
    then A1 (\) A is non-descending by Th31;
    hence thesis by SETLIM_1:def 1;
  end;
end;
