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
term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y
term TransSet = \x:set.!y:set.y iIn x -> Subq y x
const minus_SNo : set set
term - = minus_SNo
axiom SNo_minus_SNo: !x:set.SNo x -> SNo - 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 SNoL : set set
const SNoLev : set set
axiom SNoL_E: !x:set.SNo x -> !y:set.y iIn SNoL x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> y < x -> P) -> P
const SNoR : set set
axiom SNoR_E: !x:set.SNo x -> !y:set.y iIn SNoR x -> !P:prop.(SNo y -> SNoLev y iIn SNoLev x -> x < y -> P) -> P
const ordinal : set prop
axiom SNoLev_ordinal: !x:set.SNo x -> ordinal (SNoLev x)
const SNoS_ : set set
const SNo_ : set set prop
axiom SNoS_E2: !x:set.ordinal x -> !y:set.y iIn SNoS_ x -> !P:prop.(SNoLev y iIn x -> ordinal (SNoLev y) -> SNo y -> SNo_ (SNoLev y) y -> P) -> P
const mul_SNo : set set set
term * = mul_SNo
infix * 2291 2290
const SNoCut : set set set
lemma !x:set.!y:set.!z:set.!w:set.SNo x -> (!u:set.u iIn SNoS_ (SNoLev x) -> !v:set.SNo v -> !P:prop.(SNo (u * v) -> (!x2:set.x2 iIn SNoL u -> !y2:set.y2 iIn SNoL v -> (x2 * v + u * y2) (!x2:set.x2 iIn SNoR u -> !y2:set.y2 iIn SNoR v -> (x2 * v + u * y2) (!x2:set.x2 iIn SNoL u -> !y2:set.y2 iIn SNoR v -> (u * v + x2 * y2) < x2 * v + u * y2) -> (!x2:set.x2 iIn SNoR u -> !y2:set.y2 iIn SNoL v -> (u * v + x2 * y2) < x2 * v + u * y2) -> P) -> P) -> SNo y -> (!u:set.u iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * u) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL u -> (v * u + x * x2) < x * u + v * x2) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR u -> (v * u + x * x2) < x * u + v * x2) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR u -> (x * u + v * x2) < v * u + x * x2) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL u -> (x * u + v * x2) < v * u + x * x2) -> P) -> P) -> (!u:set.u iIn z -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> P) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn z) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn z) -> (!u:set.u iIn w -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> P) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn w) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn w) -> x * y = SNoCut z w -> (!u:set.u iIn SNoL x -> !v:set.SNo v -> SNo (u * v)) -> (!u:set.u iIn SNoR x -> !v:set.SNo v -> SNo (u * v)) -> (!u:set.u iIn SNoL x -> SNo (u * y)) -> (!u:set.u iIn SNoR x -> SNo (u * y)) -> (!u:set.u iIn SNoL y -> SNo (x * u)) -> (!u:set.u iIn SNoR y -> SNo (x * u)) -> SNoCutP z w -> !P:prop.(SNo (x * y) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> (u * y + x * v) < x * y + u * v) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> (u * y + x * v) < x * y + u * v) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> (x * y + u * v) (!u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> (x * y + u * v) P) -> P
lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.SNo x -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.SNo y2 -> !P:prop.(SNo (x2 * y2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoL y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoR y2 -> (z2 * y2 + x2 * w2) < x2 * y2 + z2 * w2) -> (!z2:set.z2 iIn SNoL x2 -> !w2:set.w2 iIn SNoR y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> (!z2:set.z2 iIn SNoR x2 -> !w2:set.w2 iIn SNoL y2 -> (x2 * y2 + z2 * w2) < z2 * y2 + x2 * w2) -> P) -> P) -> SNo y -> (!x2:set.x2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * x2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR x2 -> (y2 * x2 + x * z2) < x * x2 + y2 * z2) -> (!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL x2 -> (x * x2 + y2 * z2) < y2 * x2 + x * z2) -> P) -> P) -> (!x2:set.x2 iIn z -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn w -> !P:prop.(!y2:set.y2 iIn SNoL x -> !z2:set.z2 iIn SNoR y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> (!y2:set.y2 iIn SNoR x -> !z2:set.z2 iIn SNoL y -> x2 = y2 * y + x * z2 + - y2 * z2 -> P) -> P) -> (!x2:set.x2 iIn SNoL x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoR x -> !y2:set.SNo y2 -> SNo (x2 * y2)) -> (!x2:set.x2 iIn SNoL x -> SNo (x2 * y)) -> (!x2:set.x2 iIn SNoR x -> SNo (x2 * y)) -> (!x2:set.x2 iIn SNoL y -> SNo (x * x2)) -> (!x2:set.x2 iIn SNoR y -> SNo (x * x2)) -> u iIn z -> v iIn w -> (!x2:set.x2 iIn SNoS_ (SNoLev x) -> !y2:set.y2 iIn SNoS_ (SNoLev x) -> !z2:set.z2 iIn SNoS_ (SNoLev y) -> !w2:set.w2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x2 * y) -> SNo (x * z2) -> SNo (x2 * z2) -> SNo (y2 * y) -> SNo (x * w2) -> SNo (y2 * w2) -> SNo (x2 * w2) -> SNo (y2 * z2) -> (u = x2 * y + x * z2 + - x2 * z2 -> v = y2 * y + x * w2 + - y2 * w2 -> (x2 * y + x * z2 + y2 * w2) < y2 * y + x * w2 + x2 * z2 -> u < v) -> P) -> P) -> u < v
lemma !x:set.!y:set.!z:set.!w:set.!u:set.!v:set.!x2:set.!y2:set.(!z2:set.z2 iIn SNoS_ (SNoLev x) -> !w2:set.SNo w2 -> !P:prop.(SNo (z2 * w2) -> (!u2:set.u2 iIn SNoL z2 -> !v2:set.v2 iIn SNoL w2 -> (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2) -> (!u2:set.u2 iIn SNoR z2 -> !v2:set.v2 iIn SNoR w2 -> (u2 * w2 + z2 * v2) < z2 * w2 + u2 * v2) -> (!u2:set.u2 iIn SNoL z2 -> !v2:set.v2 iIn SNoR w2 -> (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2) -> (!u2:set.u2 iIn SNoR z2 -> !v2:set.v2 iIn SNoL w2 -> (z2 * w2 + u2 * v2) < u2 * w2 + z2 * v2) -> P) -> P) -> SNo y -> (!z2:set.z2 iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * z2) -> (!w2:set.w2 iIn SNoL x -> !u2:set.u2 iIn SNoL z2 -> (w2 * z2 + x * u2) < x * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoR x -> !u2:set.u2 iIn SNoR z2 -> (w2 * z2 + x * u2) < x * z2 + w2 * u2) -> (!w2:set.w2 iIn SNoL x -> !u2:set.u2 iIn SNoR z2 -> (x * z2 + w2 * u2) < w2 * z2 + x * u2) -> (!w2:set.w2 iIn SNoR x -> !u2:set.u2 iIn SNoL z2 -> (x * z2 + w2 * u2) < w2 * z2 + x * u2) -> P) -> P) -> u iIn SNoS_ (SNoLev x) -> v iIn SNoS_ (SNoLev x) -> x2 iIn SNoS_ (SNoLev y) -> y2 iIn SNoS_ (SNoLev y) -> SNo x2 -> SNo y2 -> SNo (u * x2) -> !P:prop.(SNo (u * y) -> SNo (x * x2) -> SNo (u * x2) -> SNo (v * y) -> SNo (x * y2) -> SNo (v * y2) -> SNo (u * y2) -> SNo (v * x2) -> (z = u * y + x * x2 + - u * x2 -> w = v * y + x * y2 + - v * y2 -> (u * y + x * x2 + v * y2) < v * y + x * y2 + u * x2 -> z < w) -> P) -> P
var x:set
var y:set
var z:set
var w:set
hyp SNo x
hyp !u:set.u iIn SNoS_ (SNoLev x) -> !v:set.SNo v -> !P:prop.(SNo (u * v) -> (!x2:set.x2 iIn SNoL u -> !y2:set.y2 iIn SNoL v -> (x2 * v + u * y2) (!x2:set.x2 iIn SNoR u -> !y2:set.y2 iIn SNoR v -> (x2 * v + u * y2) (!x2:set.x2 iIn SNoL u -> !y2:set.y2 iIn SNoR v -> (u * v + x2 * y2) < x2 * v + u * y2) -> (!x2:set.x2 iIn SNoR u -> !y2:set.y2 iIn SNoL v -> (u * v + x2 * y2) < x2 * v + u * y2) -> P) -> P
hyp SNo y
hyp !u:set.u iIn SNoS_ (SNoLev y) -> !P:prop.(SNo (x * u) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL u -> (v * u + x * x2) < x * u + v * x2) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR u -> (v * u + x * x2) < x * u + v * x2) -> (!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR u -> (x * u + v * x2) < v * u + x * x2) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL u -> (x * u + v * x2) < v * u + x * x2) -> P) -> P
hyp !u:set.u iIn z -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> P
hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn z
hyp !u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn z
hyp !u:set.u iIn w -> !P:prop.(!v:set.v iIn SNoL x -> !x2:set.x2 iIn SNoR y -> u = v * y + x * x2 + - v * x2 -> P) -> (!v:set.v iIn SNoR x -> !x2:set.x2 iIn SNoL y -> u = v * y + x * x2 + - v * x2 -> P) -> P
hyp !u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> u * y + x * v + - u * v iIn w
hyp !u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> u * y + x * v + - u * v iIn w
hyp x * y = SNoCut z w
hyp !u:set.u iIn SNoL x -> !v:set.SNo v -> SNo (u * v)
hyp !u:set.u iIn SNoR x -> !v:set.SNo v -> SNo (u * v)
hyp !u:set.u iIn SNoL x -> SNo (u * y)
hyp !u:set.u iIn SNoR x -> SNo (u * y)
hyp !u:set.u iIn SNoL y -> SNo (x * u)
claim (!u:set.u iIn SNoR y -> SNo (x * u)) -> !P:prop.(SNo (x * y) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoL y -> (u * y + x * v) < x * y + u * v) -> (!u:set.u iIn SNoR x -> !v:set.v iIn SNoR y -> (u * y + x * v) < x * y + u * v) -> (!u:set.u iIn SNoL x -> !v:set.v iIn SNoR y -> (x * y + u * v) (!u:set.u iIn SNoR x -> !v:set.v iIn SNoL y -> (x * y + u * v) P) -> P