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

theorem Th108:
  -1 <= r & r <= 1 implies arcsin r + arccos r = PI/2
proof
  assume
A1: -1 <= r & r <= 1;
  then -PI/2+PI/2 <= arccos r by Th99;
  then -PI/2 <= arccos r-PI/2 by XREAL_1:19;
  then
A2: --PI/2 >= -(arccos r-PI/2) by XREAL_1:24;
  arccos r <= PI/2+PI/2 by A1,Th99;
  then arccos r-PI/2 <= PI/2 by XREAL_1:20;
  then
A3: -(arccos r-PI/2) >= -PI/2 by XREAL_1:24;
  r = sin(PI/2)*cos arccos r - cos(PI/2)*sin arccos r by A1,Th91,SIN_COS:77
    .= sin(PI/2-arccos r) by COMPLEX2:3;
  then arcsin r = PI/2 - arccos r by A2,A3,Th69;
  hence thesis;
end;
