r in REAL by XREAL_0:def 1;
then [r,r] in [:REAL,REAL:] by ZFMISC_1:87;
then [r,r] in dom addreal by FUNCT_2:def 1;
then r in dom (curry addreal) by FUNCT_5:19;
hence ( (curry addreal) . r is Function-like & (curry addreal) . r is Relation-like ) by FUNCT_5:30; :: thesis: verum