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

theorem Th68:
  -1 <= r & r <= 1 implies sin arcsin r = r
proof
  assume -1 <= r & r <= 1;
  then
A1: r in [.-1,1.] by XXREAL_1:1;
  then
A2: (arcsin.r) in [.-PI/2,PI/2.] by Th62,Th63,FUNCT_1:def 3;
  thus sin arcsin r = sin.(arcsin.r) by SIN_COS:def 17
    .= ((sin|[.-PI/2,PI/2.]) qua Function).(arcsin.r) by A2,FUNCT_1:49
    .= (id [.-1,1.]).r by A1,Th63,Th64,FUNCT_1:13
    .= r by A1,FUNCT_1:18;
end;
