const In : set set prop term iIn = In infix iIn 2000 2000 const SNo : set prop const SNoLt : set set prop term < = SNoLt infix < 2020 2020 term SNoCutP = \x:set.\y:set.(!z:set.z iIn x -> SNo z) & (!z:set.z iIn y -> SNo z) & !z:set.z iIn x -> !w:set.w iIn y -> z < w const Pi : set (set set) set term setexp = \x:set.\y:set.Pi y \z:set.x const add_SNo : set set set term + = add_SNo infix + 2281 2280 axiom SNo_add_SNo: !x:set.!y:set.SNo x -> SNo y -> SNo (x + y) const SNoS_ : set set const omega : set const ap : set set set const eps_ : set set const SNoCut : set set set const Repl : set (set set) set const real : set axiom SNo_approx_real: !x:set.SNo x -> !y:set.y iIn setexp (SNoS_ omega) omega -> !z:set.z iIn setexp (SNoS_ omega) omega -> (!w:set.w iIn omega -> ap y w < x) -> (!w:set.w iIn omega -> x < ap y w + eps_ w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap y u < ap y w) -> (!w:set.w iIn omega -> x < ap z w) -> (!w:set.w iIn omega -> !u:set.u iIn w -> ap z w < ap z u) -> x = SNoCut (Repl omega (ap y)) (Repl omega (ap z)) -> x iIn real const Sigma : set (set set) set const ordsucc : set set const minus_SNo : set set term - = minus_SNo const abs_SNo : set set var x:set var y:set var z:set var w:set var u:set var v:set hyp x iIn real hyp y iIn real hyp SNo x hyp SNo y hyp SNo (x + y) hyp !x2:set.x2 iIn SNoS_ omega -> (!y2:set.y2 iIn omega -> abs_SNo (x2 + - x) < eps_ y2) -> x2 = x hyp !x2:set.x2 iIn SNoS_ omega -> (!y2:set.y2 iIn omega -> abs_SNo (x2 + - y) < eps_ y2) -> x2 = y hyp !x2:set.x2 iIn omega -> SNo (ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2)) x2) hyp !x2:set.x2 iIn omega -> SNo (ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2)) x2) hyp Sigma omega (\x2:set.ap z (ordsucc x2) + ap u (ordsucc x2)) iIn setexp (SNoS_ omega) omega hyp Sigma omega (\x2:set.ap w (ordsucc x2) + ap v (ordsucc x2)) iIn setexp (SNoS_ omega) omega hyp !x2:set.x2 iIn omega -> ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2)) x2 < x + y hyp !x2:set.x2 iIn omega -> (x + y) < ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2)) x2 + eps_ x2 hyp !x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap (Sigma omega \z2:set.ap z (ordsucc z2) + ap u (ordsucc z2)) y2 < ap (Sigma omega \z2:set.ap z (ordsucc z2) + ap u (ordsucc z2)) x2 hyp !x2:set.x2 iIn omega -> (ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2)) x2 + - eps_ x2) < x + y hyp !x2:set.x2 iIn omega -> (x + y) < ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2)) x2 hyp !x2:set.x2 iIn omega -> !y2:set.y2 iIn x2 -> ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) x2 < ap (Sigma omega \z2:set.ap w (ordsucc z2) + ap v (ordsucc z2)) y2 hyp SNoCutP (Repl omega (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap u (ordsucc x2)))) (Repl omega (ap (Sigma omega \x2:set.ap w (ordsucc x2) + ap v (ordsucc x2)))) hyp SNo (SNoCut (Repl omega (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap u (ordsucc x2)))) (Repl omega (ap (Sigma omega \x2:set.ap w (ordsucc x2) + ap v (ordsucc x2))))) hyp !x2:set.x2 iIn Repl omega (ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2))) -> x2 < SNoCut (Repl omega (ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2)))) (Repl omega (ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2)))) hyp !x2:set.x2 iIn Repl omega (ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2))) -> SNoCut (Repl omega (ap (Sigma omega \y2:set.ap z (ordsucc y2) + ap u (ordsucc y2)))) (Repl omega (ap (Sigma omega \y2:set.ap w (ordsucc y2) + ap v (ordsucc y2)))) < x2 claim x + y = SNoCut (Repl omega (ap (Sigma omega \x2:set.ap z (ordsucc x2) + ap u (ordsucc x2)))) (Repl omega (ap (Sigma omega \x2:set.ap w (ordsucc x2) + ap v (ordsucc x2)))) -> x + y iIn real