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

theorem Th77:
  -1 < r & r < 1 implies -PI/2 < arcsin r & arcsin r < PI/2
proof
  assume
A1: -1 < r & r < 1;
  then -PI/2 <= arcsin r & arcsin r <= PI/2 by Th76;
  then
  -PI/2 < arcsin r & arcsin r < PI/2 or -PI/2 = arcsin r or arcsin r = PI/
  2 by XXREAL_0:1;
  hence thesis by A1,Th7,Th68,SIN_COS:77;
end;
