( 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_3:48, ZFMISC_1:38;
hence rng (chi x,y) c= [.0 ,1.] by XBOOLE_1:1; :: according to RELAT_1:def 19 :: thesis: verum