for x0 being Real st x0 in ].0 ,PI .[ holds
diff cot ,x0 < 0

proof

let x0 be Real; :: thesis: ( x0 in ].0 ,PI .[ implies diff cot ,x0 < 0 )
assume A1: x0 in ].0 ,PI .[ ; :: thesis: diff cot ,x0 < 0
then 0 < sin . x0 by COMPTRIG:23;
then (sin . x0) ^2 > 0 by SQUARE_1:74;
then 1 / ((sin . x0) ^2 ) > 0 / ((sin . x0) ^2 ) by XREAL_1:76;
then - (1 / ((sin . x0) ^2 )) < - 0 by XREAL_1:26;
hence diff cot ,x0 < 0 by A1, Lm4; :: thesis: verum

end;

then rng (cot | ].0 ,PI .[) is open by Lm2, Th2, FDIFF_2:41;
hence dom arccot is open by FUNCT_1:55; :: thesis: verum