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

theorem Th76:
  -1 <= r & r <= 1 implies -PI/2 <= arcsin r & arcsin r <= PI/2
proof
  assume -1 <= r & r <= 1;
  then r in [.-1,1.] by XXREAL_1:1;
  then arcsin.r in rng arcsin by Th63,FUNCT_1:def 3;
  hence thesis by Th62,XXREAL_1:1;
end;
