const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y const SNo : set prop const add_SNo : set set set term + = add_SNo infix + 2281 2280 const Empty : set axiom add_SNo_0R: !x:set.SNo x -> x + Empty = x const nat_p : set prop const omega : set axiom nat_p_omega: !x:set.nat_p x -> x iIn omega axiom nat_p_SNo: !x:set.nat_p x -> SNo x const ordsucc : set set axiom SNo_1: SNo (ordsucc Empty) axiom add_SNo_In_omega: !x:set.x iIn omega -> !y:set.y iIn omega -> x + y iIn omega axiom ordsuccI1: !x:set.Subq x (ordsucc x) axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const minus_SNo : set set term - = minus_SNo axiom SNo_minus_SNo: !x:set.SNo x -> SNo - x axiom ordsuccI2: !x:set.x iIn ordsucc x axiom add_SNo_minus_L2: !x:set.!y:set.SNo x -> SNo y -> - x + x + y = y axiom add_SNo_com: !x:set.!y:set.SNo x -> SNo y -> x + y = y + x axiom ordsuccE: !x:set.!y:set.y iIn ordsucc x -> y iIn x | y = x axiom add_SNo_1_ordsucc: !x:set.x iIn omega -> x + ordsucc Empty = ordsucc x axiom add_SNo_assoc: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x + y + z = (x + y) + z axiom nat_ind: !p:set prop.p Empty -> (!x:set.nat_p x -> p x -> p (ordsucc x)) -> !x:set.nat_p x -> p x var x:set var y:set hyp y iIn omega claim SNo y -> !z:set.nat_p z -> x iIn y + z -> x iIn y | x + - y iIn z