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

theorem Th31:
  A1 is non-descending implies A1 (\) A is non-descending
proof
  assume
A1: A1 is non-descending;
  for n,m st n <= m holds (A1 (\) A).n c= (A1 (\) A).m
  proof
    let n,m;
    assume n <= m;
    then A1.n c= A1.m by A1,PROB_1:def 5;
    then A1.n \ A c= A1.m \ A by XBOOLE_1:33;
    then (A1 (\) A).n c= A1.m \ A by Def8;
    hence thesis by Def8;
  end;
  hence thesis by PROB_1:def 5;
end;
