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

theorem Th25:
  A1 is non-descending implies A (\/) A1 is non-descending
proof
  assume
A1: A1 is non-descending;
  for n,m st n <= m holds (A (\/) A1).n c= (A (\/) A1).m
  proof
    let n,m;
    assume n <= m;
    then A1.n c= A1.m by A1,PROB_1:def 5;
    then A \/ A1.n c= A \/ A1.m by XBOOLE_1:9;
    then (A (\/) A1).n c= A \/ A1.m by Def6;
    hence thesis by Def6;
  end;
  hence thesis by PROB_1:def 5;
end;
