theorem Th25:
  arctan | [.-1,1.] = (tan | [.-PI/4,PI/4.])"
proof
  set h = tan | ].-PI/2,PI/2.[;
A1: [.-PI/4,PI/4.] c= ].-PI/2,PI/2.[ by Lm7,Lm8,XXREAL_2:def 12;
  then (tan | [.-PI/4,PI/4.])" = (h | [.-PI/4,PI/4.])" by RELAT_1:74
    .= h" | (h.:[.-PI/4,PI/4.]) by RFUNCT_2:17
    .= h" | rng (h | [.-PI/4,PI/4.]) by RELAT_1:115
    .= h" | ([.-1,1.]) by A1,Th21,RELAT_1:74;
  hence thesis;
end;
