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

theorem
  -1 <= r & r <= 1 implies arccos r - arcsin(-r) = PI/2
proof
  assume
A1: -1 <= r & r <= 1;
  then
A2: arcsin r + arccos r = PI/2 + 0 by Th108;
  --1 >= -r & -r >= -1 by A1,XREAL_1:24;
  then arcsin(-r) = -arcsin(--r) by Th78
    .= arccos r - PI/2 by A2;
  hence thesis;
end;
