( 1 in [.0,1.] & 0 in [.0,1.] ) by XXREAL_1:1;
then ( rng (chi (x,y)) c= {0,1} & {0,1} c= [.0,1.] ) by FUNCT_3:39, ZFMISC_1:32;
hence rng (chi (x,y)) c= [.0,1.] by XBOOLE_1:1; :: according to RELAT_1:def 19 :: thesis: verum