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

theorem Th27:
  A1 is non-ascending implies A (\) A1 is non-descending
proof
  assume
A1: A1 is non-ascending;
  for n,m st n <= m holds (A (\) A1).n c= (A (\) A1).m
  proof
    let n,m;
    assume n <= m;
    then A1.m c= A1.n by A1,PROB_1:def 4;
    then A \ A1.n c= A \ A1.m by XBOOLE_1:34;
    then (A (\) A1).n c= A \ A1.m by Def7;
    hence thesis by Def7;
  end;
  hence thesis by PROB_1:def 5;
end;
