const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const SNo : set prop const eps_ : set set axiom SNo_eps_: !x:set.x iIn omega -> SNo (eps_ x) const add_SNo : set set set term + = add_SNo infix + 2281 2280 const nat_p : set prop const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 lemma !x:set.!y:set.x iIn omega -> y iIn omega -> x + y iIn omega -> nat_p (x + y) -> SNo (eps_ x) -> SNo (eps_ y) -> SNo (eps_ x * eps_ y) -> SNo (eps_ (x + y)) -> eps_ x * eps_ y = eps_ (x + y) var x:set var y:set hyp x iIn omega hyp y iIn omega hyp x + y iIn omega hyp nat_p (x + y) hyp SNo (eps_ x) hyp SNo (eps_ y) claim SNo (eps_ x * eps_ y) -> eps_ x * eps_ y = eps_ (x + y)