%   ORIGINAL: h4/realax/TREAL__BIJ__WELLDEF
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/realax/treal__eq0: !y2 y1 x2 x1. h4/realax/treal__eq (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) <=> h4/hreal/hreal__add x1 y2 = h4/hreal/hreal__add x2 y1
% Assm: h4/realax/TREAL__BIJ_c1: !r. h4/realax/treal__lt h4/realax/treal__0 r <=> h4/realax/treal__eq (h4/realax/treal__of__hreal (h4/realax/hreal__of__treal r)) r
% Assm: h4/realax/TREAL__EQ__REFL: !x. h4/realax/treal__eq x x
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/hreal/HREAL__ADD__SYM: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
% Assm: h4/realax/hreal__of__treal0: !y x. h4/realax/hreal__of__treal (h4/pair/_2C x y) = h4/min/_40 (\d. x = h4/hreal/hreal__add y d)
% Assm: h4/realax/HREAL__EQ__LADD: !z y x. h4/hreal/hreal__add x y = h4/hreal/hreal__add x z <=> y = z
% Assm: h4/hreal/HREAL__ADD__ASSOC: !Z Y X. h4/hreal/hreal__add X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__add X Y) Z
% Assm: h4/realax/treal__lt0: !y2 y1 x2 x1. h4/realax/treal__lt (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) <=> h4/hreal/hreal__lt (h4/hreal/hreal__add x1 y2) (h4/hreal/hreal__add x2 y1)
% Assm: h4/realax/TREAL__EQ__SYM: !y x. h4/realax/treal__eq x y <=> h4/realax/treal__eq y x
% Assm: h4/realax/TREAL__EQ__TRANS: !z y x. h4/realax/treal__eq x y /\ h4/realax/treal__eq y z ==> h4/realax/treal__eq x z
% Assm: h4/realax/treal__00: h4/realax/treal__0 = h4/pair/_2C h4/hreal/hreal__1 h4/hreal/hreal__1
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/realax/TREAL__BIJ_c0: !h. h4/realax/hreal__of__treal (h4/realax/treal__of__hreal h) = h
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/realax/treal__of__hreal0: !x. h4/realax/treal__of__hreal x = h4/pair/_2C (h4/hreal/hreal__add x h4/hreal/hreal__1) h4/hreal/hreal__1
% Assm: h4/realax/TREAL__EQ__EQUIV: !q p. h4/realax/treal__eq p q <=> h4/realax/treal__eq p = h4/realax/treal__eq q
% Assm: h4/realax/TREAL__EQ__AP: !q p. p = q ==> h4/realax/treal__eq p q
% Assm: h4/realax/TREAL__LT__TOTAL: !y x. h4/realax/treal__eq x y \/ h4/realax/treal__lt x y \/ h4/realax/treal__lt y x
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/realax/HREAL__LT__LADD: !z y x. h4/hreal/hreal__lt (h4/hreal/hreal__add x y) (h4/hreal/hreal__add x z) <=> h4/hreal/hreal__lt y z
% Assm: h4/hreal/HREAL__LT__TOTAL: !Y X. X = Y \/ h4/hreal/hreal__lt X Y \/ h4/hreal/hreal__lt Y X
% Assm: h4/hreal/HREAL__SUB__ADD: !Y X. h4/hreal/hreal__lt X Y ==> h4/hreal/hreal__add (h4/hreal/hreal__sub Y X) X = Y
% Assm: h4/realax/HREAL__LT__ADDL: !y x. h4/hreal/hreal__lt x (h4/hreal/hreal__add x y)
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/realax/treal__add0: !y2 y1 x2 x1. h4/realax/treal__add (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) = h4/pair/_2C (h4/hreal/hreal__add x1 x2) (h4/hreal/hreal__add y1 y2)
% Assm: h4/realax/TREAL__MUL__LID: !x. h4/realax/treal__eq (h4/realax/treal__mul h4/realax/treal__1 x) x
% Assm: h4/realax/TREAL__ADD__LID: !x. h4/realax/treal__eq (h4/realax/treal__add h4/realax/treal__0 x) x
% Assm: h4/realax/TREAL__MUL__LINV: !x. ~h4/realax/treal__eq x h4/realax/treal__0 ==> h4/realax/treal__eq (h4/realax/treal__mul (h4/realax/treal__inv x) x) h4/realax/treal__1
% Assm: h4/realax/TREAL__ADD__LINV: !x. h4/realax/treal__eq (h4/realax/treal__add (h4/realax/treal__neg x) x) h4/realax/treal__0
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/realax/treal__mul0: !y2 y1 x2 x1. h4/realax/treal__mul (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) = h4/pair/_2C (h4/hreal/hreal__add (h4/hreal/hreal__mul x1 x2) (h4/hreal/hreal__mul y1 y2)) (h4/hreal/hreal__add (h4/hreal/hreal__mul x1 y2) (h4/hreal/hreal__mul y1 x2))
% Assm: h4/realax/TREAL__INV__0: h4/realax/treal__eq (h4/realax/treal__inv h4/realax/treal__0) h4/realax/treal__0
% Assm: h4/realax/TREAL__10: ~h4/realax/treal__eq h4/realax/treal__1 h4/realax/treal__0
% Assm: h4/realax/treal__10: h4/realax/treal__1 = h4/pair/_2C (h4/hreal/hreal__add h4/hreal/hreal__1 h4/hreal/hreal__1) h4/hreal/hreal__1
% Assm: h4/realax/HREAL__RDISTRIB: !z y x. h4/hreal/hreal__mul (h4/hreal/hreal__add x y) z = h4/hreal/hreal__add (h4/hreal/hreal__mul x z) (h4/hreal/hreal__mul y z)
% Assm: h4/hreal/HREAL__LDISTRIB: !Z Y X. h4/hreal/hreal__mul X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__mul X Y) (h4/hreal/hreal__mul X Z)
% Assm: h4/realax/treal__inv0: !y x. h4/realax/treal__inv (h4/pair/_2C x y) = h4/bool/COND (x = y) h4/realax/treal__0 (h4/bool/COND (h4/hreal/hreal__lt y x) (h4/pair/_2C (h4/hreal/hreal__add (h4/hreal/hreal__inv (h4/hreal/hreal__sub x y)) h4/hreal/hreal__1) h4/hreal/hreal__1) (h4/pair/_2C h4/hreal/hreal__1 (h4/hreal/hreal__add (h4/hreal/hreal__inv (h4/hreal/hreal__sub y x)) h4/hreal/hreal__1)))
% Assm: h4/realax/TREAL__MUL__ASSOC: !z y x. h4/realax/treal__mul x (h4/realax/treal__mul y z) = h4/realax/treal__mul (h4/realax/treal__mul x y) z
% Assm: h4/realax/TREAL__LDISTRIB: !z y x. h4/realax/treal__mul x (h4/realax/treal__add y z) = h4/realax/treal__add (h4/realax/treal__mul x y) (h4/realax/treal__mul x z)
% Assm: h4/realax/TREAL__ADD__ASSOC: !z y x. h4/realax/treal__add x (h4/realax/treal__add y z) = h4/realax/treal__add (h4/realax/treal__add x y) z
% Assm: h4/realax/TREAL__ADD__SYM: !y x. h4/realax/treal__add x y = h4/realax/treal__add y x
% Assm: h4/realax/TREAL__MUL__SYM: !y x. h4/realax/treal__mul x y = h4/realax/treal__mul y x
% Assm: h4/realax/TREAL__LT__TRANS: !z y x. h4/realax/treal__lt x y /\ h4/realax/treal__lt y z ==> h4/realax/treal__lt x z
% Assm: h4/realax/TREAL__LT__ADD: !z y x. h4/realax/treal__lt y z ==> h4/realax/treal__lt (h4/realax/treal__add x y) (h4/realax/treal__add x z)
% Assm: h4/realax/TREAL__LT__MUL: !y x. h4/realax/treal__lt h4/realax/treal__0 x /\ h4/realax/treal__lt h4/realax/treal__0 y ==> h4/realax/treal__lt h4/realax/treal__0 (h4/realax/treal__mul x y)
% Assm: h4/realax/TREAL__LT__REFL: !x. ~h4/realax/treal__lt x x
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/realax/HREAL__LT__REFL: !x. ~h4/hreal/hreal__lt x x
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/pair/PAIR__EQ: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm: h4/hreal/HREAL__MUL__LID: !X. h4/hreal/hreal__mul h4/hreal/hreal__1 X = X
% Assm: h4/realax/treal__neg0: !y x. h4/realax/treal__neg (h4/pair/_2C x y) = h4/pair/_2C y x
% Assm: h4/hreal/HREAL__NOZERO: !Y X. ~(h4/hreal/hreal__add X Y = X)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/hreal/HREAL__LT: !Y X. h4/hreal/hreal__lt X Y <=> (?D. Y = h4/hreal/hreal__add X D)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/hreal/HREAL__MUL__LINV: !X. h4/hreal/hreal__mul (h4/hreal/hreal__inv X) X = h4/hreal/hreal__1
% Assm: h4/hreal/HREAL__MUL__SYM: !Y X. h4/hreal/hreal__mul X Y = h4/hreal/hreal__mul Y X
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/realax/HREAL__LT__ADD2: !y2 y1 x2 x1. h4/hreal/hreal__lt x1 y1 /\ h4/hreal/hreal__lt x2 y2 ==> h4/hreal/hreal__lt (h4/hreal/hreal__add x1 x2) (h4/hreal/hreal__add y1 y2)
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/realax/HREAL__LT__GT: !y x. h4/hreal/hreal__lt x y ==> ~h4/hreal/hreal__lt y x
% Assm: h4/hreal/HREAL__MUL__ASSOC: !Z Y X. h4/hreal/hreal__mul X (h4/hreal/hreal__mul Y Z) = h4/hreal/hreal__mul (h4/hreal/hreal__mul X Y) Z
% Assm: h4/hreal/hreal__tybij_c1: !r. h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm: h4/realax/HREAL__EQ__ADDL: !y x. ~(x = h4/hreal/hreal__add x y)
% Assm: h4/hreal/hreal__add0: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal0 (\w. ?x y. w = h4/hrat/hrat__add x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/hreal/HREAL__ADD__TOTAL: !Y X. X = Y \/ (?D. Y = h4/hreal/hreal__add X D) \/ (?D. X = h4/hreal/hreal__add Y D)
% Assm: h4/realax/TREAL__ISO: !i h. h4/hreal/hreal__lt h i ==> h4/realax/treal__lt (h4/realax/treal__of__hreal h) (h4/realax/treal__of__hreal i)
% Assm: h4/hreal/hreal__sub0: !Y X. h4/hreal/hreal__sub Y X = h4/hreal/hreal0 (\w. ?x. ~h4/hreal/cut X x /\ h4/hreal/cut Y (h4/hrat/hrat__add x w))
% Assm: h4/hreal/hreal__sup0: !P. h4/hreal/hreal__sup P = h4/hreal/hreal0 (\w. ?X. P X /\ h4/hreal/cut X w)
% Assm: h4/hreal/EQUAL__CUTS: !Y X. h4/hreal/cut X = h4/hreal/cut Y ==> X = Y
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/hreal/hreal__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION h4/hreal/isacut rep
% Assm: h4/hreal/hreal__tybij_c0: !a. h4/hreal/hreal0 (h4/hreal/cut a) = a
% Assm: h4/realax/HREAL__LT__NE: !y x. h4/hreal/hreal__lt x y ==> ~(x = y)
% Assm: h4/hreal/HREAL__SUP: !P. (?X. P X) /\ (?Y. !X. P X ==> h4/hreal/hreal__lt X Y) ==> (!Y. (?X. P X /\ h4/hreal/hreal__lt Y X) <=> h4/hreal/hreal__lt Y (h4/hreal/hreal__sup P))
% Assm: h4/hreal/HREAL__ADD__ISACUT: !Y X. h4/hreal/isacut (\w. ?x y. w = h4/hrat/hrat__add x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/realax/HREAL__EQ__ADDR: !y x. ~(h4/hreal/hreal__add x y = x)
% Assm: h4/hreal/CUT__NEARTOP__ADD: !e X. ?x. h4/hreal/cut X x /\ ~h4/hreal/cut X (h4/hrat/hrat__add x e)
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/hreal/CUT__NONEMPTY: !X. ?x. h4/hreal/cut X x
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/hreal/HREAL__LT__LEMMA: !Y X. h4/hreal/hreal__lt X Y ==> (?x. ~h4/hreal/cut X x /\ h4/hreal/cut Y x)
% Assm: h4/hreal/CUT__DOWN: !y x X. h4/hreal/cut X x /\ h4/hreal/hrat__lt y x ==> h4/hreal/cut X y
% Assm: h4/hreal/CUT__STRADDLE: !y x X. h4/hreal/cut X x /\ ~h4/hreal/cut X y ==> h4/hreal/hrat__lt x y
% Assm: h4/hrat/HRAT__MUL__LID: !h. h4/hrat/hrat__mul h4/hrat/hrat__1 h = h
% Assm: h4/hreal/hreal__mul0: !Y X. h4/hreal/hreal__mul X Y = h4/hreal/hreal0 (\w. ?x y. w = h4/hrat/hrat__mul x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/hrat/HRAT__MUL__ASSOC: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__mul i j) = h4/hrat/hrat__mul (h4/hrat/hrat__mul h i) j
% Assm: h4/hreal/HREAL__SUB__ISACUT: !Y X. h4/hreal/hreal__lt X Y ==> h4/hreal/isacut (\w. ?x. ~h4/hreal/cut X x /\ h4/hreal/cut Y (h4/hrat/hrat__add x w))
% Assm: h4/hreal/HRAT__LT__ADDR: !y x. h4/hreal/hrat__lt y (h4/hrat/hrat__add x y)
% Assm: h4/hreal/CUT__UBOUND: !y x X. ~h4/hreal/cut X x /\ h4/hreal/hrat__lt x y ==> ~h4/hreal/cut X y
% Assm: h4/hreal/CUT__UP: !x X. h4/hreal/cut X x ==> (?y. h4/hreal/cut X y /\ h4/hreal/hrat__lt x y)
% Assm: h4/hreal/HRAT__DOWN2: !y x. ?z. h4/hreal/hrat__lt z x /\ h4/hreal/hrat__lt z y
% Assm: h4/hreal/HRAT__LT__LADD: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add z x) (h4/hrat/hrat__add z y) <=> h4/hreal/hrat__lt x y
% Assm: h4/hreal/HRAT__LT__RADD: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add x z) (h4/hrat/hrat__add y z) <=> h4/hreal/hrat__lt x y
% Assm: h4/hrat/HRAT__ADD__ASSOC: !j i h. h4/hrat/hrat__add h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__add h i) j
% Assm: h4/hrat/HRAT__ADD__SYM: !i h. h4/hrat/hrat__add h i = h4/hrat/hrat__add i h
% Assm: h4/hreal/hrat__lt0: !y x. h4/hreal/hrat__lt x y <=> (?d. y = h4/hrat/hrat__add x d)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/hreal/HREAL__MUL__ISACUT: !Y X. h4/hreal/isacut (\w. ?x y. w = h4/hrat/hrat__mul x y /\ h4/hreal/cut X x /\ h4/hreal/cut Y y)
% Assm: h4/hreal/HRAT__LT__L1: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) h4/hrat/hrat__1 <=> h4/hreal/hrat__lt y x
% Assm: h4/hreal/HRAT__LT__LMUL1: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x y) y <=> h4/hreal/hrat__lt x h4/hrat/hrat__1
% Assm: h4/hreal/HRAT__MUL__RINV: !x. h4/hrat/hrat__mul x (h4/hrat/hrat__inv x) = h4/hrat/hrat__1
% Assm: h4/hreal/HRAT__LT__TOTAL: !y x. x = y \/ h4/hreal/hrat__lt x y \/ h4/hreal/hrat__lt y x
% Assm: h4/hrat/HRAT__LDISTRIB: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__mul h i) (h4/hrat/hrat__mul h j)
% Assm: h4/hreal/HREAL__INV__ISACUT: !X. h4/hreal/isacut (\w. ?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. h4/hreal/cut X x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))
% Assm: h4/hreal/hreal__inv0: !X. h4/hreal/hreal__inv X = h4/hreal/hreal0 (\w. ?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. h4/hreal/cut X x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))
% Assm: h4/hreal/hreal__10: h4/hreal/hreal__1 = h4/hreal/hreal0 (h4/hreal/cut__of__hrat h4/hrat/hrat__1)
% Assm: h4/hreal/CUT__NEARTOP__MUL: !u X. h4/hreal/hrat__lt h4/hrat/hrat__1 u ==> (?x. h4/hreal/cut X x /\ ~h4/hreal/cut X (h4/hrat/hrat__mul u x))
% Assm: h4/hreal/HRAT__GT__L1: !y x. h4/hreal/hrat__lt h4/hrat/hrat__1 (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) <=> h4/hreal/hrat__lt x y
% Assm: h4/hreal/HRAT__MEAN: !y x. h4/hreal/hrat__lt x y ==> (?z. h4/hreal/hrat__lt x z /\ h4/hreal/hrat__lt z y)
% Assm: h4/hreal/HRAT__LT__LMUL: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul z x) (h4/hrat/hrat__mul z y) <=> h4/hreal/hrat__lt x y
% Assm: h4/hreal/cut__of__hrat0: !x. h4/hreal/cut__of__hrat x = (\y. h4/hreal/hrat__lt y x)
% Assm: h4/hreal/HRAT__MUL__RID: !x. h4/hrat/hrat__mul x h4/hrat/hrat__1 = x
% Assm: h4/hreal/HRAT__LT__TRANS: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
% Assm: h4/hrat/HRAT__MUL__LINV: !h. h4/hrat/hrat__mul (h4/hrat/hrat__inv h) h = h4/hrat/hrat__1
% Assm: h4/hrat/HRAT__MUL__SYM: !i h. h4/hrat/hrat__mul h i = h4/hrat/hrat__mul i h
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/hreal/HREAL__SUP__ISACUT: !P. (?X. P X) /\ (?Y. !X. P X ==> h4/hreal/hreal__lt X Y) ==> h4/hreal/isacut (\w. ?X. P X /\ h4/hreal/cut X w)
% Assm: h4/hreal/hreal__lt0: !Y X. h4/hreal/hreal__lt X Y <=> ~(X = Y) /\ (!x. h4/hreal/cut X x ==> h4/hreal/cut Y x)
% Goal: !i h. h4/realax/treal__eq h i ==> h4/realax/hreal__of__treal h = h4/realax/hreal__of__treal i
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_realaxs_trealu_u_eq0]: !y2 y1 x2 x1. happ (h4/realax/treal__eq (h4/pair/_2C x1 y1)) (h4/pair/_2C x2 y2) <=> h4/hreal/hreal__add x1 y2 = h4/hreal/hreal__add x2 y1
% Assm [h4s_realaxs_TREALu_u_BIJu_c1]: !r. h4/realax/treal__lt h4/realax/treal__0 r <=> happ (h4/realax/treal__eq (h4/realax/treal__of__hreal (h4/realax/hreal__of__treal r))) r
% Assm [h4s_realaxs_TREALu_u_EQu_u_REFL]: !x. happ (h4/realax/treal__eq x) x
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_hreals_HREALu_u_ADDu_u_SYM]: !Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal__add Y X
% Assm [h4s_realaxs_hrealu_u_ofu_u_treal0]: !_0. (!x y d. happ (happ (happ _0 x) y) d <=> x = h4/hreal/hreal__add y d) ==> (!y x. h4/realax/hreal__of__treal (h4/pair/_2C x y) = h4/min/_40 (happ (happ _0 x) y))
% Assm [h4s_realaxs_HREALu_u_EQu_u_LADD]: !z y x. h4/hreal/hreal__add x y = h4/hreal/hreal__add x z <=> y = z
% Assm [h4s_hreals_HREALu_u_ADDu_u_ASSOC]: !Z Y X. h4/hreal/hreal__add X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__add X Y) Z
% Assm [h4s_realaxs_trealu_u_lt0]: !y2 y1 x2 x1. h4/realax/treal__lt (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) <=> h4/hreal/hreal__lt (h4/hreal/hreal__add x1 y2) (h4/hreal/hreal__add x2 y1)
% Assm [h4s_realaxs_TREALu_u_EQu_u_SYM]: !y x. happ (h4/realax/treal__eq x) y <=> happ (h4/realax/treal__eq y) x
% Assm [h4s_realaxs_TREALu_u_EQu_u_TRANS]: !z y x. happ (h4/realax/treal__eq x) y /\ happ (h4/realax/treal__eq y) z ==> happ (h4/realax/treal__eq x) z
% Assm [h4s_realaxs_trealu_u_00]: h4/realax/treal__0 = h4/pair/_2C h4/hreal/hreal__1 h4/hreal/hreal__1
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_realaxs_TREALu_u_BIJu_c0]: !h. h4/realax/hreal__of__treal (h4/realax/treal__of__hreal h) = h
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_realaxs_trealu_u_ofu_u_hreal0]: !x. h4/realax/treal__of__hreal x = h4/pair/_2C (h4/hreal/hreal__add x h4/hreal/hreal__1) h4/hreal/hreal__1
% Assm [h4s_realaxs_TREALu_u_EQu_u_EQUIV]: !q p. happ (h4/realax/treal__eq p) q <=> h4/realax/treal__eq p = h4/realax/treal__eq q
% Assm [h4s_realaxs_TREALu_u_EQu_u_AP]: !q p. p = q ==> happ (h4/realax/treal__eq p) q
% Assm [h4s_realaxs_TREALu_u_LTu_u_TOTAL]: !y x. happ (h4/realax/treal__eq x) y \/ h4/realax/treal__lt x y \/ h4/realax/treal__lt y x
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_realaxs_HREALu_u_LTu_u_LADD]: !z y x. h4/hreal/hreal__lt (h4/hreal/hreal__add x y) (h4/hreal/hreal__add x z) <=> h4/hreal/hreal__lt y z
% Assm [h4s_hreals_HREALu_u_LTu_u_TOTAL]: !Y X. X = Y \/ h4/hreal/hreal__lt X Y \/ h4/hreal/hreal__lt Y X
% Assm [h4s_hreals_HREALu_u_SUBu_u_ADD]: !Y X. h4/hreal/hreal__lt X Y ==> h4/hreal/hreal__add (h4/hreal/hreal__sub Y X) X = Y
% Assm [h4s_realaxs_HREALu_u_LTu_u_ADDL]: !y x. h4/hreal/hreal__lt x (h4/hreal/hreal__add x y)
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_realaxs_trealu_u_add0]: !y2 y1 x2 x1. h4/realax/treal__add (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) = h4/pair/_2C (h4/hreal/hreal__add x1 x2) (h4/hreal/hreal__add y1 y2)
% Assm [h4s_realaxs_TREALu_u_MULu_u_LID]: !x. happ (h4/realax/treal__eq (h4/realax/treal__mul h4/realax/treal__1 x)) x
% Assm [h4s_realaxs_TREALu_u_ADDu_u_LID]: !x. happ (h4/realax/treal__eq (h4/realax/treal__add h4/realax/treal__0 x)) x
% Assm [h4s_realaxs_TREALu_u_MULu_u_LINV]: !x. ~happ (h4/realax/treal__eq x) h4/realax/treal__0 ==> happ (h4/realax/treal__eq (h4/realax/treal__mul (h4/realax/treal__inv x) x)) h4/realax/treal__1
% Assm [h4s_realaxs_TREALu_u_ADDu_u_LINV]: !x. happ (h4/realax/treal__eq (h4/realax/treal__add (h4/realax/treal__neg x) x)) h4/realax/treal__0
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_realaxs_trealu_u_mul0]: !y2 y1 x2 x1. h4/realax/treal__mul (h4/pair/_2C x1 y1) (h4/pair/_2C x2 y2) = h4/pair/_2C (h4/hreal/hreal__add (h4/hreal/hreal__mul x1 x2) (h4/hreal/hreal__mul y1 y2)) (h4/hreal/hreal__add (h4/hreal/hreal__mul x1 y2) (h4/hreal/hreal__mul y1 x2))
% Assm [h4s_realaxs_TREALu_u_INVu_u_0]: happ (h4/realax/treal__eq (h4/realax/treal__inv h4/realax/treal__0)) h4/realax/treal__0
% Assm [h4s_realaxs_TREALu_u_10]: ~happ (h4/realax/treal__eq h4/realax/treal__1) h4/realax/treal__0
% Assm [h4s_realaxs_trealu_u_10]: h4/realax/treal__1 = h4/pair/_2C (h4/hreal/hreal__add h4/hreal/hreal__1 h4/hreal/hreal__1) h4/hreal/hreal__1
% Assm [h4s_realaxs_HREALu_u_RDISTRIB]: !z y x. h4/hreal/hreal__mul (h4/hreal/hreal__add x y) z = h4/hreal/hreal__add (h4/hreal/hreal__mul x z) (h4/hreal/hreal__mul y z)
% Assm [h4s_hreals_HREALu_u_LDISTRIB]: !Z Y X. h4/hreal/hreal__mul X (h4/hreal/hreal__add Y Z) = h4/hreal/hreal__add (h4/hreal/hreal__mul X Y) (h4/hreal/hreal__mul X Z)
% Assm [h4s_realaxs_trealu_u_inv0]: !y x. ?v. (v <=> x = y) /\ h4/realax/treal__inv (h4/pair/_2C x y) = h4/bool/COND v h4/realax/treal__0 (h4/bool/COND (h4/hreal/hreal__lt y x) (h4/pair/_2C (h4/hreal/hreal__add (h4/hreal/hreal__inv (h4/hreal/hreal__sub x y)) h4/hreal/hreal__1) h4/hreal/hreal__1) (h4/pair/_2C h4/hreal/hreal__1 (h4/hreal/hreal__add (h4/hreal/hreal__inv (h4/hreal/hreal__sub y x)) h4/hreal/hreal__1)))
% Assm [h4s_realaxs_TREALu_u_MULu_u_ASSOC]: !z y x. h4/realax/treal__mul x (h4/realax/treal__mul y z) = h4/realax/treal__mul (h4/realax/treal__mul x y) z
% Assm [h4s_realaxs_TREALu_u_LDISTRIB]: !z y x. h4/realax/treal__mul x (h4/realax/treal__add y z) = h4/realax/treal__add (h4/realax/treal__mul x y) (h4/realax/treal__mul x z)
% Assm [h4s_realaxs_TREALu_u_ADDu_u_ASSOC]: !z y x. h4/realax/treal__add x (h4/realax/treal__add y z) = h4/realax/treal__add (h4/realax/treal__add x y) z
% Assm [h4s_realaxs_TREALu_u_ADDu_u_SYM]: !y x. h4/realax/treal__add x y = h4/realax/treal__add y x
% Assm [h4s_realaxs_TREALu_u_MULu_u_SYM]: !y x. h4/realax/treal__mul x y = h4/realax/treal__mul y x
% Assm [h4s_realaxs_TREALu_u_LTu_u_TRANS]: !z y x. h4/realax/treal__lt x y /\ h4/realax/treal__lt y z ==> h4/realax/treal__lt x z
% Assm [h4s_realaxs_TREALu_u_LTu_u_ADD]: !z y x. h4/realax/treal__lt y z ==> h4/realax/treal__lt (h4/realax/treal__add x y) (h4/realax/treal__add x z)
% Assm [h4s_realaxs_TREALu_u_LTu_u_MUL]: !y x. h4/realax/treal__lt h4/realax/treal__0 x /\ h4/realax/treal__lt h4/realax/treal__0 y ==> h4/realax/treal__lt h4/realax/treal__0 (h4/realax/treal__mul x y)
% Assm [h4s_realaxs_TREALu_u_LTu_u_REFL]: !x. ~h4/realax/treal__lt x x
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_realaxs_HREALu_u_LTu_u_REFL]: !x. ~h4/hreal/hreal__lt x x
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_pairs_PAIRu_u_EQ]: !y x b a. h4/pair/_2C x y = h4/pair/_2C a b <=> x = a /\ y = b
% Assm [h4s_hreals_HREALu_u_MULu_u_LID]: !X. h4/hreal/hreal__mul h4/hreal/hreal__1 X = X
% Assm [h4s_realaxs_trealu_u_neg0]: !y x. h4/realax/treal__neg (h4/pair/_2C x y) = h4/pair/_2C y x
% Assm [h4s_hreals_HREALu_u_NOZERO]: !Y X. ~(h4/hreal/hreal__add X Y = X)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_hreals_HREALu_u_LT]: !Y X. h4/hreal/hreal__lt X Y <=> (?D. Y = h4/hreal/hreal__add X D)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_hreals_HREALu_u_MULu_u_LINV]: !X. h4/hreal/hreal__mul (h4/hreal/hreal__inv X) X = h4/hreal/hreal__1
% Assm [h4s_hreals_HREALu_u_MULu_u_SYM]: !Y X. h4/hreal/hreal__mul X Y = h4/hreal/hreal__mul Y X
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_realaxs_HREALu_u_LTu_u_ADD2]: !y2 y1 x2 x1. h4/hreal/hreal__lt x1 y1 /\ h4/hreal/hreal__lt x2 y2 ==> h4/hreal/hreal__lt (h4/hreal/hreal__add x1 x2) (h4/hreal/hreal__add y1 y2)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_realaxs_HREALu_u_LTu_u_GT]: !y x. h4/hreal/hreal__lt x y ==> ~h4/hreal/hreal__lt y x
% Assm [h4s_hreals_HREALu_u_MULu_u_ASSOC]: !Z Y X. h4/hreal/hreal__mul X (h4/hreal/hreal__mul Y Z) = h4/hreal/hreal__mul (h4/hreal/hreal__mul X Y) Z
% Assm [h4s_hreals_hrealu_u_tybiju_c1]: !r. happ h4/hreal/isacut r <=> h4/hreal/cut (h4/hreal/hreal0 r) = r
% Assm [h4s_realaxs_HREALu_u_EQu_u_ADDL]: !y x. ~(x = h4/hreal/hreal__add x y)
% Assm [h4s_hreals_hrealu_u_add0]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__add x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. h4/hreal/hreal__add X Y = h4/hreal/hreal0 (happ (happ _0 X) Y))
% Assm [h4s_hreals_HREALu_u_ADDu_u_TOTAL]: !Y X. X = Y \/ (?D. Y = h4/hreal/hreal__add X D) \/ (?D. X = h4/hreal/hreal__add Y D)
% Assm [h4s_realaxs_TREALu_u_ISO]: !i h. h4/hreal/hreal__lt h i ==> h4/realax/treal__lt (h4/realax/treal__of__hreal h) (h4/realax/treal__of__hreal i)
% Assm [h4s_hreals_hrealu_u_sub0]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x. ~happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) (h4/hrat/hrat__add x w))) ==> (!Y X. h4/hreal/hreal__sub Y X = h4/hreal/hreal0 (happ (happ _0 X) Y))
% Assm [h4s_hreals_hrealu_u_sup0]: !_0. (!P w. happ (happ _0 P) w <=> (?X. happ P X /\ happ (h4/hreal/cut X) w)) ==> (!P. h4/hreal/hreal__sup P = h4/hreal/hreal0 (happ _0 P))
% Assm [h4s_hreals_EQUALu_u_CUTS]: !Y X. h4/hreal/cut X = h4/hreal/cut Y ==> X = Y
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_hreals_hrealu_u_TYu_u_DEF]: ?rep. h4/bool/TYPE__DEFINITION h4/hreal/isacut rep
% Assm [h4s_hreals_hrealu_u_tybiju_c0]: !a. h4/hreal/hreal0 (h4/hreal/cut a) = a
% Assm [h4s_realaxs_HREALu_u_LTu_u_NE]: !y x. h4/hreal/hreal__lt x y ==> ~(x = y)
% Assm [h4s_hreals_HREALu_u_SUP]: !P. (?X. happ P X) /\ (?Y. !X. happ P X ==> h4/hreal/hreal__lt X Y) ==> (!Y. (?X. happ P X /\ h4/hreal/hreal__lt Y X) <=> h4/hreal/hreal__lt Y (h4/hreal/hreal__sup P))
% Assm [h4s_hreals_HREALu_u_ADDu_u_ISACUT]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__add x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. happ h4/hreal/isacut (happ (happ _0 X) Y))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_realaxs_HREALu_u_EQu_u_ADDR]: !y x. ~(h4/hreal/hreal__add x y = x)
% Assm [h4s_hreals_CUTu_u_NEARTOPu_u_ADD]: !e X. ?x. happ (h4/hreal/cut X) x /\ ~happ (h4/hreal/cut X) (h4/hrat/hrat__add x e)
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_hreals_CUTu_u_NONEMPTY]: !X. ?x. happ (h4/hreal/cut X) x
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_hreals_HREALu_u_LTu_u_LEMMA]: !Y X. h4/hreal/hreal__lt X Y ==> (?x. ~happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) x)
% Assm [h4s_hreals_CUTu_u_DOWN]: !y x X. happ (h4/hreal/cut X) x /\ h4/hreal/hrat__lt y x ==> happ (h4/hreal/cut X) y
% Assm [h4s_hreals_CUTu_u_STRADDLE]: !y x X. happ (h4/hreal/cut X) x /\ ~happ (h4/hreal/cut X) y ==> h4/hreal/hrat__lt x y
% Assm [h4s_hrats_HRATu_u_MULu_u_LID]: !h. h4/hrat/hrat__mul h4/hrat/hrat__1 h = h
% Assm [h4s_hreals_hrealu_u_mul0]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__mul x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. h4/hreal/hreal__mul X Y = h4/hreal/hreal0 (happ (happ _0 X) Y))
% Assm [h4s_hrats_HRATu_u_MULu_u_ASSOC]: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__mul i j) = h4/hrat/hrat__mul (h4/hrat/hrat__mul h i) j
% Assm [h4s_hreals_HREALu_u_SUBu_u_ISACUT]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x. ~happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) (h4/hrat/hrat__add x w))) ==> (!Y X. h4/hreal/hreal__lt X Y ==> happ h4/hreal/isacut (happ (happ _0 X) Y))
% Assm [h4s_hreals_HRATu_u_LTu_u_ADDR]: !y x. h4/hreal/hrat__lt y (h4/hrat/hrat__add x y)
% Assm [h4s_hreals_CUTu_u_UBOUND]: !y x X. ~happ (h4/hreal/cut X) x /\ h4/hreal/hrat__lt x y ==> ~happ (h4/hreal/cut X) y
% Assm [h4s_hreals_CUTu_u_UP]: !x X. happ (h4/hreal/cut X) x ==> (?y. happ (h4/hreal/cut X) y /\ h4/hreal/hrat__lt x y)
% Assm [h4s_hreals_HRATu_u_DOWN2]: !y x. ?z. h4/hreal/hrat__lt z x /\ h4/hreal/hrat__lt z y
% Assm [h4s_hreals_HRATu_u_LTu_u_LADD]: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add z x) (h4/hrat/hrat__add z y) <=> h4/hreal/hrat__lt x y
% Assm [h4s_hreals_HRATu_u_LTu_u_RADD]: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__add x z) (h4/hrat/hrat__add y z) <=> h4/hreal/hrat__lt x y
% Assm [h4s_hrats_HRATu_u_ADDu_u_ASSOC]: !j i h. h4/hrat/hrat__add h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__add h i) j
% Assm [h4s_hrats_HRATu_u_ADDu_u_SYM]: !i h. h4/hrat/hrat__add h i = h4/hrat/hrat__add i h
% Assm [h4s_hreals_hratu_u_lt0]: !y x. h4/hreal/hrat__lt x y <=> (?d. y = h4/hrat/hrat__add x d)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_hreals_HREALu_u_MULu_u_ISACUT]: !_0. (!X Y w. happ (happ (happ _0 X) Y) w <=> (?x y. w = h4/hrat/hrat__mul x y /\ happ (h4/hreal/cut X) x /\ happ (h4/hreal/cut Y) y)) ==> (!Y X. happ h4/hreal/isacut (happ (happ _0 X) Y))
% Assm [h4s_hreals_HRATu_u_LTu_u_L1]: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) h4/hrat/hrat__1 <=> h4/hreal/hrat__lt y x
% Assm [h4s_hreals_HRATu_u_LTu_u_LMUL1]: !y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul x y) y <=> h4/hreal/hrat__lt x h4/hrat/hrat__1
% Assm [h4s_hreals_HRATu_u_MULu_u_RINV]: !x. h4/hrat/hrat__mul x (h4/hrat/hrat__inv x) = h4/hrat/hrat__1
% Assm [h4s_hreals_HRATu_u_LTu_u_TOTAL]: !y x. x = y \/ h4/hreal/hrat__lt x y \/ h4/hreal/hrat__lt y x
% Assm [h4s_hrats_HRATu_u_LDISTRIB]: !j i h. h4/hrat/hrat__mul h (h4/hrat/hrat__add i j) = h4/hrat/hrat__add (h4/hrat/hrat__mul h i) (h4/hrat/hrat__mul h j)
% Assm [h4s_hreals_HREALu_u_INVu_u_ISACUT]: !_0. (!X w. happ (happ _0 X) w <=> (?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. happ (h4/hreal/cut X) x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))) ==> (!X. happ h4/hreal/isacut (happ _0 X))
% Assm [h4s_hreals_hrealu_u_inv0]: !_0. (!X w. happ (happ _0 X) w <=> (?d. h4/hreal/hrat__lt d h4/hrat/hrat__1 /\ (!x. happ (h4/hreal/cut X) x ==> h4/hreal/hrat__lt (h4/hrat/hrat__mul w x) d))) ==> (!X. h4/hreal/hreal__inv X = h4/hreal/hreal0 (happ _0 X))
% Assm [h4s_hreals_hrealu_u_10]: h4/hreal/hreal__1 = h4/hreal/hreal0 (h4/hreal/cut__of__hrat h4/hrat/hrat__1)
% Assm [h4s_hreals_CUTu_u_NEARTOPu_u_MUL]: !u X. h4/hreal/hrat__lt h4/hrat/hrat__1 u ==> (?x. happ (h4/hreal/cut X) x /\ ~happ (h4/hreal/cut X) (h4/hrat/hrat__mul u x))
% Assm [h4s_hreals_HRATu_u_GTu_u_L1]: !y x. h4/hreal/hrat__lt h4/hrat/hrat__1 (h4/hrat/hrat__mul (h4/hrat/hrat__inv x) y) <=> h4/hreal/hrat__lt x y
% Assm [h4s_hreals_HRATu_u_MEAN]: !y x. h4/hreal/hrat__lt x y ==> (?z. h4/hreal/hrat__lt x z /\ h4/hreal/hrat__lt z y)
% Assm [h4s_hreals_HRATu_u_LTu_u_LMUL]: !z y x. h4/hreal/hrat__lt (h4/hrat/hrat__mul z x) (h4/hrat/hrat__mul z y) <=> h4/hreal/hrat__lt x y
% Assm [h4s_hreals_cutu_u_ofu_u_hrat0]: !x x'. happ (h4/hreal/cut__of__hrat x) x' <=> h4/hreal/hrat__lt x' x
% Assm [h4s_hreals_HRATu_u_MULu_u_RID]: !x. h4/hrat/hrat__mul x h4/hrat/hrat__1 = x
% Assm [h4s_hreals_HRATu_u_LTu_u_TRANS]: !z y x. h4/hreal/hrat__lt x y /\ h4/hreal/hrat__lt y z ==> h4/hreal/hrat__lt x z
% Assm [h4s_hrats_HRATu_u_MULu_u_LINV]: !h. h4/hrat/hrat__mul (h4/hrat/hrat__inv h) h = h4/hrat/hrat__1
% Assm [h4s_hrats_HRATu_u_MULu_u_SYM]: !i h. h4/hrat/hrat__mul h i = h4/hrat/hrat__mul i h
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_hreals_HREALu_u_SUPu_u_ISACUT]: !_0. (!P w. happ (happ _0 P) w <=> (?X. happ P X /\ happ (h4/hreal/cut X) w)) ==> (!P. (?X. happ P X) /\ (?Y. !X. happ P X ==> h4/hreal/hreal__lt X Y) ==> happ h4/hreal/isacut (happ _0 P))
% Assm [h4s_hreals_hrealu_u_lt0]: !Y X. h4/hreal/hreal__lt X Y <=> ~(X = Y) /\ (!x. happ (h4/hreal/cut X) x ==> happ (h4/hreal/cut Y) x)
% Goal: !i h. happ (h4/realax/treal__eq h) i ==> h4/realax/hreal__of__treal h = h4/realax/hreal__of__treal i
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1424102,TV_Q1424098]: ![V_f, V_g]: (![V_x]: s(TV_Q1424098,happ(s(t_fun(TV_Q1424102,TV_Q1424098),V_f),s(TV_Q1424102,V_x))) = s(TV_Q1424098,happ(s(t_fun(TV_Q1424102,TV_Q1424098),V_g),s(TV_Q1424102,V_x))) => s(t_fun(TV_Q1424102,TV_Q1424098),V_f) = s(t_fun(TV_Q1424102,TV_Q1424098),V_g))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_realaxs_trealu_u_eq0, axiom, ![V_y2, V_y1, V_x2, V_x1]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y1))))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y2)))))) <=> s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y2))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y1))))).
fof(ah4s_realaxs_TREALu_u_BIJu_c1, axiom, ![V_r]: s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_r))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_ofu_u_hreal(s(t_h4s_hreals_hreal,h4s_realaxs_hrealu_u_ofu_u_treal(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_r))))))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_r)))).
fof(ah4s_realaxs_TREALu_u_EQu_u_REFL, axiom, ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_hreals_HREALu_u_ADDu_u_SYM, axiom, ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X)))).
fof(ah4s_realaxs_hrealu_u_ofu_u_treal0, axiom, ![V_uu_0]: (![V_x, V_y, V_d]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_x))),s(t_h4s_hreals_hreal,V_y))),s(t_h4s_hreals_hreal,V_d)))) <=> s(t_h4s_hreals_hreal,V_x) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_d)))) => ![V_y, V_x]: s(t_h4s_hreals_hreal,h4s_realaxs_hrealu_u_ofu_u_treal(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))))) = s(t_h4s_hreals_hreal,h4s_mins_u_40(s(t_fun(t_h4s_hreals_hreal,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_x))),s(t_h4s_hreals_hreal,V_y))))))).
fof(ah4s_realaxs_HREALu_u_EQu_u_LADD, axiom, ![V_z, V_y, V_x]: (s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_z))) <=> s(t_h4s_hreals_hreal,V_y) = s(t_h4s_hreals_hreal,V_z))).
fof(ah4s_hreals_HREALu_u_ADDu_u_ASSOC, axiom, ![V_Z, V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_Z))))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hreals_hreal,V_Z)))).
fof(ah4s_realaxs_trealu_u_lt0, axiom, ![V_y2, V_y1, V_x2, V_x1]: s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y2))))) = s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y2))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y1)))))).
fof(ah4s_realaxs_TREALu_u_EQu_u_SYM, axiom, ![V_y, V_x]: s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))) = s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x)))).
fof(ah4s_realaxs_TREALu_u_EQu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))) & p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))))).
fof(ah4s_realaxs_trealu_u_00, axiom, s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1)))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_realaxs_TREALu_u_BIJu_c0, axiom, ![V_h]: s(t_h4s_hreals_hreal,h4s_realaxs_hrealu_u_ofu_u_treal(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_ofu_u_hreal(s(t_h4s_hreals_hreal,V_h))))) = s(t_h4s_hreals_hreal,V_h)).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_realaxs_trealu_u_ofu_u_hreal0, axiom, ![V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_ofu_u_hreal(s(t_h4s_hreals_hreal,V_x))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1)))).
fof(ah4s_realaxs_TREALu_u_EQu_u_EQUIV, axiom, ![V_q, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_p))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_q)))) <=> s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_p))) = s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_q))))).
fof(ah4s_realaxs_TREALu_u_EQu_u_AP, axiom, ![V_q, V_p]: (s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_p) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_q) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_p))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_q)))))).
fof(ah4s_realaxs_TREALu_u_LTu_u_TOTAL, axiom, ![V_y, V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))) | (p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))) | p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_realaxs_HREALu_u_LTu_u_LADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_z))))) = s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_z)))).
fof(ah4s_hreals_HREALu_u_LTu_u_TOTAL, axiom, ![V_Y, V_X]: (s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y) | (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) | p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X))))))).
fof(ah4s_hreals_HREALu_u_SUBu_u_ADD, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) => s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sub(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_X))) = s(t_h4s_hreals_hreal,V_Y))).
fof(ah4s_realaxs_HREALu_u_LTu_u_ADDL, axiom, ![V_y, V_x]: p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))))))).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_realaxs_trealu_u_add0, axiom, ![V_y2, V_y1, V_x2, V_x1]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y2))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_x2))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_y1),s(t_h4s_hreals_hreal,V_y2)))))).
fof(ah4s_realaxs_TREALu_u_MULu_u_LID, axiom, ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_1),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))).
fof(ah4s_realaxs_TREALu_u_ADDu_u_LID, axiom, ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))).
fof(ah4s_realaxs_TREALu_u_MULu_u_LINV, axiom, ![V_x]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_inv(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_1)))))).
fof(ah4s_realaxs_TREALu_u_ADDu_u_LINV, axiom, ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_neg(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x))))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_realaxs_trealu_u_mul0, axiom, ![V_y2, V_y1, V_x2, V_x1]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y2))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_x2))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_y1),s(t_h4s_hreals_hreal,V_y2))))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y2))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_y1),s(t_h4s_hreals_hreal,V_x2)))))))).
fof(ah4s_realaxs_TREALu_u_INVu_u_0, axiom, p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_inv(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0))))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0))))).
fof(ah4s_realaxs_TREALu_u_10, axiom, ~ (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0)))))).
fof(ah4s_realaxs_trealu_u_10, axiom, s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_1) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1)))).
fof(ah4s_realaxs_HREALu_u_RDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))),s(t_h4s_hreals_hreal,V_z))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_z))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_z)))))).
fof(ah4s_hreals_HREALu_u_LDISTRIB, axiom, ![V_Z, V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_Z))))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Z)))))).
fof(ah4s_realaxs_trealu_u_inv0, axiom, ![V_y, V_x]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_hreals_hreal,V_x) = s(t_h4s_hreals_hreal,V_y)) & s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_inv(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_bools_cond(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_x))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_inv(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sub(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_inv(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sub(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_x))))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1))))))))))).
fof(ah4s_realaxs_TREALu_u_MULu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))).
fof(ah4s_realaxs_TREALu_u_LDISTRIB, axiom, ![V_z, V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))))).
fof(ah4s_realaxs_TREALu_u_ADDu_u_ASSOC, axiom, ![V_z, V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))).
fof(ah4s_realaxs_TREALu_u_ADDu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x)))).
fof(ah4s_realaxs_TREALu_u_MULu_u_SYM, axiom, ![V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x)))).
fof(ah4s_realaxs_TREALu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))) & p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z))))) => p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))))).
fof(ah4s_realaxs_TREALu_u_LTu_u_ADD, axiom, ![V_z, V_y, V_x]: (p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))) => p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_add(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_z)))))))).
fof(ah4s_realaxs_TREALu_u_LTu_u_MUL, axiom, ![V_y, V_x]: ((p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x)))) & p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y))))) => p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_0),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_mul(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_y)))))))).
fof(ah4s_realaxs_TREALu_u_LTu_u_REFL, axiom, ![V_x]: ~ (p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_x)))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_realaxs_HREALu_u_LTu_u_REFL, axiom, ![V_x]: ~ (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_x)))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_pairs_PAIRu_u_EQ, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x, V_b, V_a]: (s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_a),s(TV_u_27b,V_b))) <=> (s(TV_u_27a,V_x) = s(TV_u_27a,V_a) & s(TV_u_27b,V_y) = s(TV_u_27b,V_b)))).
fof(ah4s_hreals_HREALu_u_MULu_u_LID, axiom, ![V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1),s(t_h4s_hreals_hreal,V_X))) = s(t_h4s_hreals_hreal,V_X)).
fof(ah4s_realaxs_trealu_u_neg0, axiom, ![V_y, V_x]: s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_neg(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))))) = s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_pairs_u_2c(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_x)))).
fof(ah4s_hreals_HREALu_u_NOZERO, axiom, ![V_Y, V_X]: ~ (s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,V_X))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_hreals_HREALu_u_LT, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) <=> ?[V_D]: s(t_h4s_hreals_hreal,V_Y) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_D))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_hreals_HREALu_u_MULu_u_LINV, axiom, ![V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_inv(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_X))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1)).
fof(ah4s_hreals_HREALu_u_MULu_u_SYM, axiom, ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_realaxs_HREALu_u_LTu_u_ADD2, axiom, ![V_y2, V_y1, V_x2, V_x1]: ((p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_y1)))) & p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x2),s(t_h4s_hreals_hreal,V_y2))))) => p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x1),s(t_h4s_hreals_hreal,V_x2))),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_y1),s(t_h4s_hreals_hreal,V_y2)))))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_realaxs_HREALu_u_LTu_u_GT, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y)))) => ~ (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_y),s(t_h4s_hreals_hreal,V_x))))))).
fof(ah4s_hreals_HREALu_u_MULu_u_ASSOC, axiom, ![V_Z, V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_Z))))) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hreals_hreal,V_Z)))).
fof(ah4s_hreals_hrealu_u_tybiju_c1, axiom, ![V_r]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),V_r)))) <=> s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))))) = s(t_fun(t_h4s_hrats_hrat,t_bool),V_r))).
fof(ah4s_realaxs_HREALu_u_EQu_u_ADDL, axiom, ![V_y, V_x]: ~ (s(t_h4s_hreals_hreal,V_x) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))))).
fof(ah4s_hreals_hrealu_u_add0, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))))))).
fof(ah4s_hreals_HREALu_u_ADDu_u_TOTAL, axiom, ![V_Y, V_X]: (s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y) | (?[V_D]: s(t_h4s_hreals_hreal,V_Y) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_D))) | ?[V_D]: s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_D)))))).
fof(ah4s_realaxs_TREALu_u_ISO, axiom, ![V_i, V_h]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_h),s(t_h4s_hreals_hreal,V_i)))) => p(s(t_bool,h4s_realaxs_trealu_u_lt(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_ofu_u_hreal(s(t_h4s_hreals_hreal,V_h))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),h4s_realaxs_trealu_u_ofu_u_hreal(s(t_h4s_hreals_hreal,V_i)))))))).
fof(ah4s_hreals_hrealu_u_sub0, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_w)))))))) => ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sub(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))))))).
fof(ah4s_hreals_hrealu_u_sup0, axiom, ![V_uu_0]: (![V_P, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))))) => ![V_P]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sup(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))))))).
fof(ah4s_hreals_EQUALu_u_CUTS, axiom, ![V_Y, V_X]: (s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))) = s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))) => s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_hreals_hrealu_u_TYu_u_DEF, axiom, ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_rep))))).
fof(ah4s_hreals_hrealu_u_tybiju_c0, axiom, ![V_a]: s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_a))))) = s(t_h4s_hreals_hreal,V_a)).
fof(ah4s_realaxs_HREALu_u_LTu_u_NE, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y)))) => ~ (s(t_h4s_hreals_hreal,V_x) = s(t_h4s_hreals_hreal,V_y)))).
fof(ah4s_hreals_HREALu_u_SUP, axiom, ![V_P]: ((?[V_X]: p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & ?[V_Y]: ![V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) => p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))))) => ![V_Y]: (?[V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,V_X))))) <=> p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_Y),s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_sup(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))))))))).
fof(ah4s_hreals_HREALu_u_ADDu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y)))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_realaxs_HREALu_u_EQu_u_ADDR, axiom, ![V_y, V_x]: ~ (s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_add(s(t_h4s_hreals_hreal,V_x),s(t_h4s_hreals_hreal,V_y))) = s(t_h4s_hreals_hreal,V_x))).
fof(ah4s_hreals_CUTu_u_NEARTOPu_u_ADD, axiom, ![V_e, V_X]: ?[V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_e))))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_hreals_CUTu_u_NONEMPTY, axiom, ![V_X]: ?[V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_hreals_HREALu_u_LTu_u_LEMMA, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) => ?[V_x]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_x))))))).
fof(ah4s_hreals_CUTu_u_DOWN, axiom, ![V_y, V_x, V_X]: ((p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))) => p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y)))))).
fof(ah4s_hreals_CUTu_u_STRADDLE, axiom, ![V_y, V_x, V_X]: ((p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y)))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))))).
fof(ah4s_hrats_HRATu_u_MULu_u_LID, axiom, ![V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1),s(t_h4s_hrats_hrat,V_h))) = s(t_h4s_hrats_hrat,V_h)).
fof(ah4s_hreals_hrealu_u_mul0, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_mul(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))))))).
fof(ah4s_hrats_HRATu_u_MULu_u_ASSOC, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,V_j)))).
fof(ah4s_hreals_HREALu_u_SUBu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_w)))))))) => ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))))))))).
fof(ah4s_hreals_HRATu_u_LTu_u_ADDR, axiom, ![V_y, V_x]: p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_CUTu_u_UBOUND, axiom, ![V_y, V_x, V_X]: ((~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x))))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))) => ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_CUTu_u_UP, axiom, ![V_x, V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => ?[V_y]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_HRATu_u_DOWN2, axiom, ![V_y, V_x]: ?[V_z]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y)))))).
fof(ah4s_hreals_HRATu_u_LTu_u_LADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hreals_HRATu_u_LTu_u_RADD, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hrats_HRATu_u_ADDu_u_ASSOC, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,V_j)))).
fof(ah4s_hrats_HRATu_u_ADDu_u_SYM, axiom, ![V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_h)))).
fof(ah4s_hreals_hratu_u_lt0, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) <=> ?[V_d]: s(t_h4s_hrats_hrat,V_y) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_d))))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_hreals_HREALu_u_MULu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_Y, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_x, V_y]: (s(t_h4s_hrats_hrat,V_w) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))) & (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_y))))))) => ![V_Y, V_X]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool))),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hreals_hreal,V_Y)))))))).
fof(ah4s_hreals_HRATu_u_LTu_u_L1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_y))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x)))).
fof(ah4s_hreals_HRATu_u_LTu_u_LMUL1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y))),s(t_h4s_hrats_hrat,V_y))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))).
fof(ah4s_hreals_HRATu_u_MULu_u_RINV, axiom, ![V_x]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_x))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)).
fof(ah4s_hreals_HRATu_u_LTu_u_TOTAL, axiom, ![V_y, V_x]: (s(t_h4s_hrats_hrat,V_x) = s(t_h4s_hrats_hrat,V_y) | (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) | p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_x))))))).
fof(ah4s_hrats_HRATu_u_LDISTRIB, axiom, ![V_j, V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_j))))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_add(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_j)))))).
fof(ah4s_hreals_HREALu_u_INVu_u_ISACUT, axiom, ![V_uu_0]: (![V_X, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_d]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_d),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_w),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_d))))))) => ![V_X]: p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X)))))))).
fof(ah4s_hreals_hrealu_u_inv0, axiom, ![V_uu_0]: (![V_X, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_d]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_d),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_w),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_d))))))) => ![V_X]: s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_inv(s(t_h4s_hreals_hreal,V_X))) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_h4s_hreals_hreal,t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_h4s_hreals_hreal,V_X))))))).
fof(ah4s_hreals_hrealu_u_10, axiom, s(t_h4s_hreals_hreal,h4s_hreals_hrealu_u_1) = s(t_h4s_hreals_hreal,h4s_hreals_hreal0(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cutu_u_ofu_u_hrat(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)))))).
fof(ah4s_hreals_CUTu_u_NEARTOPu_u_MUL, axiom, ![V_u, V_X]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1),s(t_h4s_hrats_hrat,V_u)))) => ?[V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) & ~ (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_u),s(t_h4s_hrats_hrat,V_x)))))))))).
fof(ah4s_hreals_HRATu_u_GTu_u_L1, axiom, ![V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_y))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hreals_HRATu_u_MEAN, axiom, ![V_y, V_x]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) => ?[V_z]: (p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y))))))).
fof(ah4s_hreals_HRATu_u_LTu_u_LMUL, axiom, ![V_z, V_y, V_x]: s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_z),s(t_h4s_hrats_hrat,V_y))))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))).
fof(ah4s_hreals_cutu_u_ofu_u_hrat0, axiom, ![V_x, V_xi_]: s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cutu_u_ofu_u_hrat(s(t_h4s_hrats_hrat,V_x))),s(t_h4s_hrats_hrat,V_xi_))) = s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_xi_),s(t_h4s_hrats_hrat,V_x)))).
fof(ah4s_hreals_HRATu_u_MULu_u_RID, axiom, ![V_x]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1))) = s(t_h4s_hrats_hrat,V_x)).
fof(ah4s_hreals_HRATu_u_LTu_u_TRANS, axiom, ![V_z, V_y, V_x]: ((p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_y)))) & p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_y),s(t_h4s_hrats_hrat,V_z))))) => p(s(t_bool,h4s_hreals_hratu_u_lt(s(t_h4s_hrats_hrat,V_x),s(t_h4s_hrats_hrat,V_z)))))).
fof(ah4s_hrats_HRATu_u_MULu_u_LINV, axiom, ![V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_inv(s(t_h4s_hrats_hrat,V_h))),s(t_h4s_hrats_hrat,V_h))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_1)).
fof(ah4s_hrats_HRATu_u_MULu_u_SYM, axiom, ![V_i, V_h]: s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_h),s(t_h4s_hrats_hrat,V_i))) = s(t_h4s_hrats_hrat,h4s_hrats_hratu_u_mul(s(t_h4s_hrats_hrat,V_i),s(t_h4s_hrats_hrat,V_h)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_hreals_HREALu_u_SUPu_u_ISACUT, axiom, ![V_uu_0]: (![V_P, V_w]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))),s(t_h4s_hrats_hrat,V_w)))) <=> ?[V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_w)))))) => ![V_P]: ((?[V_X]: p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) & ?[V_Y]: ![V_X]: (p(s(t_bool,happ(s(t_fun(t_h4s_hreals_hreal,t_bool),V_P),s(t_h4s_hreals_hreal,V_X)))) => p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))))) => p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_hrats_hrat,t_bool),t_bool),h4s_hreals_isacut),s(t_fun(t_h4s_hrats_hrat,t_bool),happ(s(t_fun(t_fun(t_h4s_hreals_hreal,t_bool),t_fun(t_h4s_hrats_hrat,t_bool)),V_uu_0),s(t_fun(t_h4s_hreals_hreal,t_bool),V_P))))))))).
fof(ah4s_hreals_hrealu_u_lt0, axiom, ![V_Y, V_X]: (p(s(t_bool,h4s_hreals_hrealu_u_lt(s(t_h4s_hreals_hreal,V_X),s(t_h4s_hreals_hreal,V_Y)))) <=> (~ (s(t_h4s_hreals_hreal,V_X) = s(t_h4s_hreals_hreal,V_Y)) & ![V_x]: (p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_X))),s(t_h4s_hrats_hrat,V_x)))) => p(s(t_bool,happ(s(t_fun(t_h4s_hrats_hrat,t_bool),h4s_hreals_cut(s(t_h4s_hreals_hreal,V_Y))),s(t_h4s_hrats_hrat,V_x)))))))).
fof(ch4s_realaxs_TREALu_u_BIJu_u_WELLDEF, conjecture, ![V_i, V_h]: (p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),t_bool),h4s_realaxs_trealu_u_eq(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_h))),s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_i)))) => s(t_h4s_hreals_hreal,h4s_realaxs_hrealu_u_ofu_u_treal(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_h))) = s(t_h4s_hreals_hreal,h4s_realaxs_hrealu_u_ofu_u_treal(s(t_h4s_pairs_prod(t_h4s_hreals_hreal,t_h4s_hreals_hreal),V_i))))).
