:: XXREAL_1 semantic presentation begin scheme :: XXREAL_1:sch 1 Conti{ P1[ ($#m1_hidden :::"set"::: ) ], P2[ ($#m1_hidden :::"set"::: ) ] } : (Bool "ex" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool P1[(Set (Var "r"))])) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) ")" ) & (Bool "(" "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool P2[(Set (Var "r"))])) "holds" (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) ")" )) provided (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool P1[(Set (Var "r"))]) & (Bool P2[(Set (Var "s"))])) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) proof end; begin definitionlet "r", "s" be ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) ; func :::"[.":::"r" "," "s":::".]"::: -> ($#m1_hidden :::"set"::: ) equals :: XXREAL_1:def 1 "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool "r" ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) "s") ")" ) "}" ; func :::"[.":::"r" "," "s":::".["::: -> ($#m1_hidden :::"set"::: ) equals :: XXREAL_1:def 2 "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool "r" ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) "s") ")" ) "}" ; func :::"].":::"r" "," "s":::".]"::: -> ($#m1_hidden :::"set"::: ) equals :: XXREAL_1:def 3 "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool "r" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) "s") ")" ) "}" ; func :::"].":::"r" "," "s":::".["::: -> ($#m1_hidden :::"set"::: ) equals :: XXREAL_1:def 4 "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool "r" ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) "s") ")" ) "}" ; end; :: deftheorem defines :::"[."::: XXREAL_1:def 1 : (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) ")" ) "}" )); :: deftheorem defines :::"[."::: XXREAL_1:def 2 : (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) ")" ) "}" )); :: deftheorem defines :::"]."::: XXREAL_1:def 3 : (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) ")" ) "}" )); :: deftheorem defines :::"]."::: XXREAL_1:def 4 : (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ) : (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "a"))) & (Bool (Set (Var "a")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) ")" ) "}" )); theorem :: XXREAL_1:1 (Bool "for" (Set (Var "t")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) "iff" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) ")" ) ")" )) ; theorem :: XXREAL_1:2 (Bool "for" (Set (Var "t")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) "iff" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) ")" ) ")" )) ; theorem :: XXREAL_1:3 (Bool "for" (Set (Var "t")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) "iff" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) ")" ) ")" )) ; theorem :: XXREAL_1:4 (Bool "for" (Set (Var "t")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "t")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) "iff" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t"))) & (Bool (Set (Var "t")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) ")" ) ")" )) ; registrationlet "r", "s" be ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k1_xxreal_1 :::"[."::: ) "r" "," "s" ($#k1_xxreal_1 :::".]"::: ) ) -> ($#v2_membered :::"ext-real-membered"::: ) ; cluster (Set ($#k2_xxreal_1 :::"[."::: ) "r" "," "s" ($#k2_xxreal_1 :::".["::: ) ) -> ($#v2_membered :::"ext-real-membered"::: ) ; cluster (Set ($#k3_xxreal_1 :::"]."::: ) "r" "," "s" ($#k3_xxreal_1 :::".]"::: ) ) -> ($#v2_membered :::"ext-real-membered"::: ) ; cluster (Set ($#k4_xxreal_1 :::"]."::: ) "r" "," "s" ($#k4_xxreal_1 :::".["::: ) ) -> ($#v2_membered :::"ext-real-membered"::: ) ; end; theorem :: XXREAL_1:5 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ))) ; theorem :: XXREAL_1:6 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ))) ; theorem :: XXREAL_1:7 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ))) ; theorem :: XXREAL_1:8 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ))) ; theorem :: XXREAL_1:9 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) "or" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ))) ; theorem :: XXREAL_1:10 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) "or" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) ")" ) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" ))) ; theorem :: XXREAL_1:11 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" "not" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) "or" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))) ")" ) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ))) ; theorem :: XXREAL_1:12 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))) ")" ))) ; theorem :: XXREAL_1:13 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) ")" ))) ; theorem :: XXREAL_1:14 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))) ")" ))) ; theorem :: XXREAL_1:15 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) ")" ))) ; theorem :: XXREAL_1:16 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "p"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "q"))) ")" ))) ; theorem :: XXREAL_1:17 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "r")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:18 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "r")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:19 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "r")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:20 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "r")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; registrationlet "r" be ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k1_xxreal_1 :::"[."::: ) "r" "," "r" ($#k1_xxreal_1 :::".]"::: ) ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; cluster (Set ($#k2_xxreal_1 :::"[."::: ) "r" "," "r" ($#k2_xxreal_1 :::".["::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; cluster (Set ($#k3_xxreal_1 :::"]."::: ) "r" "," "r" ($#k3_xxreal_1 :::".]"::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; cluster (Set ($#k4_xxreal_1 :::"]."::: ) "r" "," "r" ($#k4_xxreal_1 :::".["::: ) ) -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: XXREAL_1:21 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:22 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:23 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:24 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:25 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:26 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "q")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:27 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "q")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:28 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "q")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:29 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "q")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:30 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; theorem :: XXREAL_1:31 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; theorem :: XXREAL_1:32 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; theorem :: XXREAL_1:33 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; theorem :: XXREAL_1:34 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:35 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:36 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:37 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:38 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:39 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:40 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:41 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:42 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:43 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:44 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:45 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:46 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:47 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:48 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:49 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:50 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:51 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:52 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:53 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:54 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:55 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:56 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:57 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:58 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:59 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:60 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:61 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:62 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:63 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:64 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:65 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) ")" )) ; theorem :: XXREAL_1:66 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Var "s"))) ")" )) ; theorem :: XXREAL_1:67 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Var "s"))) ")" )) ; theorem :: XXREAL_1:68 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Var "s"))) ")" )) ; theorem :: XXREAL_1:69 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "r"))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Var "s"))) ")" )) ; theorem :: XXREAL_1:70 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:71 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:72 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:73 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:74 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:75 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:76 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:77 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:78 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:79 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:80 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:81 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:82 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r2_xboole_0 :::"c<"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) & (Bool (Bool "not" (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))))) "holds" (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) ; theorem :: XXREAL_1:83 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:84 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:85 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool "(" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )) ")" )) ; theorem :: XXREAL_1:86 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:87 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:88 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:89 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:90 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:91 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:92 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:93 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:94 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:95 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:96 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:97 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:98 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:99 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:100 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:101 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:102 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:103 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:104 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:105 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:106 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:107 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:108 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:109 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:110 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:111 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:112 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:113 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:114 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:115 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:116 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:117 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:118 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:119 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:120 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:121 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:122 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:123 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:124 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:125 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:126 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:127 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool "not" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; begin theorem :: XXREAL_1:128 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:129 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:130 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:131 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:132 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:133 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:134 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:135 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:136 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:137 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:138 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:139 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:140 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:141 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:142 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:143 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:144 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:145 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:146 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:147 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:148 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:149 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:150 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:151 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:152 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:153 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:154 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:155 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:156 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:157 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:158 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:159 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:160 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:161 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:162 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:163 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:164 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set "(" ($#k3_xxreal_0 :::"min"::: ) "(" (Set (Var "r")) "," (Set (Var "p")) ")" ")" ) "," (Set "(" ($#k4_xxreal_0 :::"max"::: ) "(" (Set (Var "s")) "," (Set (Var "q")) ")" ")" ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:165 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:166 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:167 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:168 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:169 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:170 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:171 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:172 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:173 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:174 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:175 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:176 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:177 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:178 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:179 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set "(" (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:180 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set "(" (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:181 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set "(" (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:182 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:183 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:184 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:185 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:186 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:187 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:188 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:189 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:190 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:191 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:192 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:193 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:194 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:195 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:196 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:197 (Bool "for" (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:198 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:199 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:200 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set "(" (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "r")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:201 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:202 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:203 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:204 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:205 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Bool "not" (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:206 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Bool "not" (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:207 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Bool "not" (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:208 (Bool "for" (Set (Var "s")) "," (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Bool "not" (Set (Var "s")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))))) ; begin theorem :: XXREAL_1:209 (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k7_numbers :::"ExtREAL"::: ) )) ; theorem :: XXREAL_1:210 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:211 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:212 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:213 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:214 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:215 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:216 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:217 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))) ; theorem :: XXREAL_1:218 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Set (Var "q")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "q")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:219 (Bool "for" (Set (Var "q")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "q")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:220 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:221 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; begin theorem :: XXREAL_1:222 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "q")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:223 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool "not" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "t")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:224 (Bool (Set ($#k1_numbers :::"REAL"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ; theorem :: XXREAL_1:225 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: XXREAL_1:226 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: XXREAL_1:227 (Bool "for" (Set (Var "q")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) ; theorem :: XXREAL_1:228 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool (Set (Var "q")) ($#r2_hidden :::"in"::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_numbers :::"REAL"::: ) ))) ; registrationlet "p", "q" be ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k4_xxreal_1 :::"]."::: ) "p" "," "q" ($#k4_xxreal_1 :::".["::: ) ) -> ($#v3_membered :::"real-membered"::: ) ; end; registrationlet "p" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "q" be ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k2_xxreal_1 :::"[."::: ) "p" "," "q" ($#k2_xxreal_1 :::".["::: ) ) -> ($#v3_membered :::"real-membered"::: ) ; end; registrationlet "q" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "p" be ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k3_xxreal_1 :::"]."::: ) "p" "," "q" ($#k3_xxreal_1 :::".]"::: ) ) -> ($#v3_membered :::"real-membered"::: ) ; end; registrationlet "p", "q" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set ($#k1_xxreal_1 :::"[."::: ) "p" "," "q" ($#k1_xxreal_1 :::".]"::: ) ) -> ($#v3_membered :::"real-membered"::: ) ; end; theorem :: XXREAL_1:229 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Real":::) : (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) "}" )) ; theorem :: XXREAL_1:230 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Real":::) : (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "g"))) "}" )) ; theorem :: XXREAL_1:231 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Real":::) : (Bool (Set (Var "g")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) "}" )) ; theorem :: XXREAL_1:232 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) "{" (Set (Var "g")) where g "is" ($#m1_subset_1 :::"Real":::) : (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "g"))) "}" )) ; theorem :: XXREAL_1:233 (Bool "for" (Set (Var "u")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "u")) ($#k4_xxreal_1 :::".["::: ) )) "iff" (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "u"))) ")" ))) ; theorem :: XXREAL_1:234 (Bool "for" (Set (Var "u")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "u")) ($#k3_xxreal_1 :::".]"::: ) )) "iff" (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "u"))) ")" ))) ; theorem :: XXREAL_1:235 (Bool "for" (Set (Var "u")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "u")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) "iff" (Bool (Set (Var "u")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x"))) ")" ))) ; theorem :: XXREAL_1:236 (Bool "for" (Set (Var "u")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "x")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "u")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) "iff" (Bool (Set (Var "u")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "x"))) ")" ))) ; theorem :: XXREAL_1:237 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:238 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:239 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:240 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:241 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:242 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:243 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:244 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:245 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:246 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:247 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:248 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:249 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:250 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:251 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:252 (Bool "for" (Set (Var "p")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:253 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:254 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:255 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:256 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:257 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:258 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:259 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:260 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:261 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:262 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:263 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:264 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:265 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:266 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:267 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:268 (Bool "for" (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_tarski :::"c="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:269 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "b")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "a")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:270 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "b")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "b")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:271 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "a")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "a")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "a")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:272 (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "a")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "b")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "b")) "," (Set (Var "a")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:273 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:274 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:275 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:276 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:277 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:278 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:279 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:280 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "r")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p")))) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:281 (Bool "for" (Set (Var "t")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:282 (Bool "for" (Set (Var "t")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:283 (Bool "for" (Set (Var "t")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:284 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:285 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:286 (Bool "for" (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:287 (Bool "for" (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:288 (Bool "for" (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:289 (Bool "for" (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:290 (Bool "for" (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:291 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:292 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:293 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:294 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:295 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:296 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:297 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:298 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:299 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:300 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:301 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:302 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:303 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:304 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:305 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:306 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:307 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:308 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:309 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:310 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:311 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:312 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:313 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:314 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:315 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:316 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:317 (Bool "for" (Set (Var "r")) "," (Set (Var "q")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:318 (Bool "for" (Set (Var "r")) "," (Set (Var "q")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:319 (Bool "for" (Set (Var "r")) "," (Set (Var "q")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:320 (Bool "for" (Set (Var "r")) "," (Set (Var "q")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:321 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:322 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:323 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:324 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:325 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:326 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:327 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:328 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:329 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:330 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:331 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:332 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:333 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:334 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:335 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:336 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:337 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:338 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:339 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:340 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:341 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:342 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:343 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:344 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:345 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:346 (Bool "for" (Set (Var "q")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:347 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:348 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:349 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:350 (Bool "for" (Set (Var "q")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:351 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: XXREAL_1:352 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "q")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "q")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "q")) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: XXREAL_1:353 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:354 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:355 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:356 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:357 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:358 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:359 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:360 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:361 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:362 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:363 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:364 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:365 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:366 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:367 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:368 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:369 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:370 (Bool "for" (Set (Var "r")) "," (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:371 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: XXREAL_1:372 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) ))))) ; theorem :: XXREAL_1:373 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:374 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:375 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:376 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:377 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:378 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:379 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:380 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:381 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:382 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:383 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:384 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:385 (Bool "for" (Set (Var "p")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:386 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ))))) ; theorem :: XXREAL_1:387 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:388 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:389 (Bool "for" (Set (Var "p")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:390 (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:391 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:392 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:393 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:394 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:395 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:396 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:397 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:398 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:399 (Bool "for" (Set (Var "s")) "," (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_numbers :::"REAL"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "p")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:400 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "p")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "p")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ))))) ; theorem :: XXREAL_1:401 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:402 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:403 (Bool "for" (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:404 (Bool "for" (Set (Var "r")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:405 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:406 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:407 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:408 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:409 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:410 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:411 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:412 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k6_subset_1 :::"\"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "s")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:413 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:414 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k3_xxreal_1 :::".]"::: ) ))) ; theorem :: XXREAL_1:415 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "t")) ($#k4_xxreal_1 :::".["::: ) ))) ; theorem :: XXREAL_1:416 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k2_xxreal_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:417 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:418 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "," (Set (Var "t")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))) "holds" (Bool (Set (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#k3_xboole_0 :::"/\"::: ) (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "s")) "," (Set (Var "t")) ($#k1_xxreal_1 :::".]"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) ))) ; theorem :: XXREAL_1:419 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k2_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:420 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:421 (Bool "for" (Set (Var "s")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:422 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:423 (Bool "for" (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) ) "," (Set (Var "s")) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "s")) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:424 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k2_tarski :::"{"::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:425 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) )))) ; theorem :: XXREAL_1:426 (Bool "for" (Set (Var "r")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k1_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) )))) ; theorem :: XXREAL_1:427 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k2_xxreal_1 :::"[."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k2_xxreal_1 :::".["::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k1_tarski :::"{"::: ) (Set (Var "r")) ($#k1_tarski :::"}"::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) )))) ; theorem :: XXREAL_1:428 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set ($#k3_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k3_xxreal_1 :::".]"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set ($#k4_xxreal_1 :::"]."::: ) (Set (Var "r")) "," (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k4_xxreal_1 :::".["::: ) ) ($#k2_xboole_0 :::"\/"::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#k1_tarski :::"}"::: ) )))) ;