theorem Th47:
  arctan|[.-1,1.] is increasing
proof
A1: [.-1,1.] = tan.:[.-PI/4,PI/4.] by Th21,RELAT_1:115;
  [.-PI/4,PI/4.] c= ].-PI/2,PI/2.[ by Lm7,Lm8,XXREAL_2:def 12;
  hence thesis by A1,Th45,RELAT_1:123,RFUNCT_2:28;
end;
