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

theorem Th69:
  -PI/2 <= r & r <= PI/2 implies arcsin sin r = r
proof
A1: dom (sin|[.-PI/2,PI/2.]) = [.-PI/2,PI/2.] by RELAT_1:62,SIN_COS:24;
  assume -PI/2 <= r & r <= PI/2;
  then
A2: r in [.-PI/2,PI/2.] by XXREAL_1:1;
  thus arcsin sin r = arcsin.(sin.r) by SIN_COS:def 17
    .= arcsin.((sin|[.-PI/2,PI/2.]).r) by A2,FUNCT_1:49
    .= (id [.-PI/2,PI/2.]).r by A2,A1,Th66,FUNCT_1:13
    .= r by A2,FUNCT_1:18;
end;
