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

theorem Th101:
  -1 <= r & r <= 1 implies arccos r = PI - arccos(-r)
proof
  assume -1 <= r & r <= 1;
  then
A1: --1 >= -r & -r >= -1 by XREAL_1:24;
  then 0+arccos(-r) <= PI by Th99;
  then
A2: 0 <= PI-arccos(-r) by XREAL_1:19;
  0 <= arccos(-r) by A1,Th99;
  then PI+0 <= PI+arccos(-r) by XREAL_1:6;
  then
A3: PI-arccos(-r) <= PI by XREAL_1:20;
  r = (-1)*(-r)
    .= cos PI * cos arccos(-r) + sin PI * sin arccos(-r) by A1,Th91,SIN_COS:77
    .= cos(PI-arccos(-r)) by COMPLEX2:3;
  hence thesis by A2,A3,Th92;
end;
