reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem
  arccos|[.-1,1.] is decreasing
proof
  set f = cos | [.0,PI.];
  f.:[.0,PI.] = [.-1,1.] & (f|[.0,PI.])"|(f.:[.0,PI.]) is decreasing by Th49,
COMPTRIG:25,FCONT_3:10,RELAT_1:129;
  hence thesis by RELAT_1:72;
end;
