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

theorem
  arcsin|[.-1,1.] is increasing
proof
  set f = sin | [.-PI/2,PI/2.];
  f.:[.-PI/2,PI/2.] = [.-1,1.] & (f|[.-PI/2,PI/2.])"|(f.:[.-PI/2,PI/2.])
  is increasing by Th45,COMPTRIG:23,FCONT_3:9,RELAT_1:129;
  hence thesis by RELAT_1:72;
end;
