let X be non empty set ; for A being set holds
( chi (+infty,A,X) = Xchi (A,X) & chi (-infty,A,X) = - (Xchi (A,X)) )
let A be set ; ( chi (+infty,A,X) = Xchi (A,X) & chi (-infty,A,X) = - (Xchi (A,X)) )
for x being Element of X holds (chi (+infty,A,X)) . x = (Xchi (A,X)) . x
hence
chi (+infty,A,X) = Xchi (A,X)
by FUNCT_2:def 8; chi (-infty,A,X) = - (Xchi (A,X))
for x being Element of X holds (chi (-infty,A,X)) . x = (- (Xchi (A,X))) . x
hence
chi (-infty,A,X) = - (Xchi (A,X))
by FUNCT_2:def 8; verum