let f be Function; :: thesis: ( f is nonpair-yielding implies not rng f is with_pair )
assume A1: for x being set st x in dom f holds
not f . x is pair ; :: according to FACIRC_1:def 3 :: thesis: not rng f is with_pair
let y be pair set ; :: according to FACIRC_1:def 2 :: thesis: not y in rng f
assume y in rng f ; :: thesis: contradiction
then ex x being set st
( x in dom f & y = f . x ) by FUNCT_1:def 3;
hence contradiction by A1; :: thesis: verum