theorem Th27:
  for A being non-empty OSAlgebra of S holds A is monotone iff for
  o1,o2 st o1 <= o2 holds Den(o1,A) c= Den(o2,A)
proof
  let A be non-empty OSAlgebra of S;
  hereby
    assume
A1: A is monotone;
    let o1,o2;
    assume o1 <= o2;
    then Den(o2,A)|Args(o1,A) = Den(o1,A) by A1;
    hence Den(o1,A) c= Den(o2,A) by RELAT_1:59;
  end;
  assume
A2: for o1,o2 st o1 <= o2 holds Den(o1,A) c= Den(o2,A);
  let o1,o2 such that
A3: o1 <= o2;
 dom Den(o1,A) = Args(o1,A) by FUNCT_2:def 1;
  hence Den(o2,A)|Args(o1,A) = Den(o1,A)|Args(o1,A) by A2,A3,GRFUNC_1:27
    .= Den(o1,A);
end;
