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

theorem Th92:
  0 <= r & r <= PI implies arccos cos r = r
proof
A1: dom (cos|[.0,PI.]) = [.0,PI.] by RELAT_1:62,SIN_COS:24;
  assume 0 <= r & r <= PI;
  then
A2: r in [.0,PI.] by XXREAL_1:1;
  thus arccos cos r = arccos.(cos.r) by SIN_COS:def 19
    .= arccos.((cos|[.0,PI.]).r) by A2,FUNCT_1:49
    .= (id [.0,PI.]).r by A2,A1,Th89,FUNCT_1:13
    .= r by A2,FUNCT_1:18;
end;
