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

theorem Th78:
  -1 <= r & r <= 1 implies arcsin r = -arcsin(-r)
proof
  assume -1 <= r & r <= 1;
  then
A1: --1 >= -r & -r >= -1 by XREAL_1:24;
  then arcsin(-r) <= PI/2 by Th76;
  then
A2: -PI/2 <= -arcsin(-r) by XREAL_1:24;
  -PI/2 <= arcsin(-r) by A1,Th76;
  then
A3: -arcsin(-r) <= --PI/2 by XREAL_1:24;
  r = 0-1*(-r)
    .= sin 0 * cos arcsin(-r) - cos 0 * sin arcsin(-r) by A1,Th68,SIN_COS:31
    .= sin(0-arcsin(-r)) by COMPLEX2:3;
  hence thesis by A2,A3,Th69;
end;
