const In : set set prop term iIn = In infix iIn 2000 2000 const omega : set const int : set const mul_SNo : set set set term * = mul_SNo infix * 2291 2290 const eps_ : set set term diadic_rational_p = \x:set.?y:set.y iIn omega & ?z:set.z iIn int & x = eps_ y * z term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const nat_p : set prop const ordsucc : set set const Empty : set axiom nat_2: nat_p (ordsucc (ordsucc Empty)) axiom omega_nat_p: !x:set.x iIn omega -> nat_p x const exp_SNo_nat : set set set axiom nat_exp_SNo_nat: !x:set.nat_p x -> !y:set.nat_p y -> nat_p (exp_SNo_nat x y) axiom nat_p_omega: !x:set.nat_p x -> x iIn omega axiom Subq_omega_int: Subq omega int const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.z iIn omega -> SNo (eps_ z) -> w iIn int -> x = eps_ z * w -> u iIn omega -> SNo (eps_ u) -> v iIn int -> y = eps_ u * v -> SNo (eps_ u * v) -> exp_SNo_nat (ordsucc (ordsucc Empty)) u iIn int -> exp_SNo_nat (ordsucc (ordsucc Empty)) u * w iIn int -> exp_SNo_nat (ordsucc (ordsucc Empty)) z iIn int -> exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v iIn int & x + y = eps_ (z + u) * (exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v) var x:set var y:set var z:set var w:set var u:set var v:set hyp z iIn omega hyp SNo (eps_ z) hyp w iIn int hyp x = eps_ z * w hyp u iIn omega hyp SNo (eps_ u) hyp v iIn int hyp y = eps_ u * v hyp SNo (eps_ u * v) hyp exp_SNo_nat (ordsucc (ordsucc Empty)) u iIn int claim exp_SNo_nat (ordsucc (ordsucc Empty)) u * w iIn int -> exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v iIn int & x + y = eps_ (z + u) * (exp_SNo_nat (ordsucc (ordsucc Empty)) u * w + exp_SNo_nat (ordsucc (ordsucc Empty)) z * v)