:: PSCOMP_1 semantic presentation begin begin definitionlet "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; mode RealMap of "T" is ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set ($#k1_numbers :::"REAL"::: ) ); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T")) bbbadV1_FUNCT_2((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set ($#k1_numbers :::"REAL"::: ) )) ($#v1_valued_0 :::"complex-valued"::: ) ($#v2_valued_0 :::"ext-real-valued"::: ) ($#v3_valued_0 :::"real-valued"::: ) ($#v1_comseq_2 :::"bounded"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set ($#k1_numbers :::"REAL"::: ) ))))); end; definitionlet "T" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "f" be ($#m1_subset_1 :::"RealMap":::) "of" (Set (Const "T")); func :::"lower_bound"::: "f" -> ($#m1_subset_1 :::"Real":::) equals :: PSCOMP_1:def 1 (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" "f" ($#k7_relset_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); func :::"upper_bound"::: "f" -> ($#m1_subset_1 :::"Real":::) equals :: PSCOMP_1:def 2 (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" "f" ($#k7_relset_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ")" )); end; :: deftheorem defines :::"lower_bound"::: PSCOMP_1:def 1 : (Bool "for" (Set (Var "T")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k5_seq_4 :::"lower_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ")" ))))); :: deftheorem defines :::"upper_bound"::: PSCOMP_1:def 2 : (Bool "for" (Set (Var "T")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_seq_4 :::"upper_bound"::: ) (Set "(" (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) ")" ))))); theorem :: PSCOMP_1:1 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" bbbadV2_SEQ_2() ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set (Var "f"))))))) ; theorem :: PSCOMP_1:2 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" bbbadV2_SEQ_2() ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool "(" "for" (Set (Var "t")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "t"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s"))) ")" )) "holds" (Bool (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "s")))))) ; theorem :: PSCOMP_1:3 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "r"))) ")" ) & (Bool "(" "for" (Set (Var "t")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Set (Var "t"))) ")" )) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">="::: ) (Set (Var "t"))) ")" )) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set (Var "f"))))))) ; theorem :: PSCOMP_1:4 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" bbbadV1_SEQ_2() ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set (Var "f"))))))) ; theorem :: PSCOMP_1:5 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" bbbadV1_SEQ_2() ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "t")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) ")" )) "holds" (Bool (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t")))))) ; theorem :: PSCOMP_1:6 (Bool "for" (Set (Var "r")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) ")" ) & (Bool "(" "for" (Set (Var "t")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool "(" "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "T")) "holds" (Bool (Set (Set (Var "f")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) ")" )) "holds" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "t"))) ")" )) "holds" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set (Var "f"))))))) ; theorem :: PSCOMP_1:7 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "f")) "being" ($#v1_comseq_2 :::"bounded"::: ) ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "holds" (Bool (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set (Var "f"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set (Var "f")))))) ; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "f" be ($#m1_subset_1 :::"RealMap":::) "of" (Set (Const "T")); attr "f" is :::"continuous"::: means :: PSCOMP_1:def 3 (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v2_rcomp_1 :::"closed"::: ) )) "holds" (Bool (Set "f" ($#k8_relset_1 :::"""::: ) (Set (Var "Y"))) "is" ($#v4_pre_topc :::"closed"::: ) )); end; :: deftheorem defines :::"continuous"::: PSCOMP_1:def 3 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v2_rcomp_1 :::"closed"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set (Var "Y"))) "is" ($#v4_pre_topc :::"closed"::: ) )) ")" ))); registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T")) bbbadV1_FUNCT_2((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set ($#k1_numbers :::"REAL"::: ) )) ($#v1_valued_0 :::"complex-valued"::: ) ($#v2_valued_0 :::"ext-real-valued"::: ) ($#v3_valued_0 :::"real-valued"::: ) ($#v1_pscomp_1 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set ($#k1_numbers :::"REAL"::: ) ))))); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Const "T")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#v4_relat_1 :::"-defined"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) bbbadV1_FUNCT_2((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "," (Set ($#k1_numbers :::"REAL"::: ) )) ($#v1_valued_0 :::"complex-valued"::: ) ($#v2_valued_0 :::"ext-real-valued"::: ) ($#v3_valued_0 :::"real-valued"::: ) ($#v1_pscomp_1 :::"continuous"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "," (Set ($#k1_numbers :::"REAL"::: ) ))))); end; theorem :: PSCOMP_1:8 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v3_rcomp_1 :::"open"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set (Var "Y"))) "is" ($#v3_pre_topc :::"open"::: ) )) ")" ))) ; theorem :: PSCOMP_1:9 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set ($#k32_valued_1 :::"-"::: ) (Set (Var "f"))) "is" ($#v1_pscomp_1 :::"continuous"::: ) ))) ; theorem :: PSCOMP_1:10 (Bool "for" (Set (Var "r3")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set (Set (Var "r3")) ($#k9_valued_1 :::"+"::: ) (Set (Var "f"))) "is" ($#v1_pscomp_1 :::"continuous"::: ) )))) ; theorem :: PSCOMP_1:11 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) & (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f")))))) "holds" (Bool (Set ($#k7_measure6 :::"Inv"::: ) (Set (Var "f"))) "is" ($#v1_pscomp_1 :::"continuous"::: ) ))) ; theorem :: PSCOMP_1:12 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) & (Bool (Set (Var "R")) "is" ($#v1_measure6 :::"open"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_measure6 :::"""::: ) (Set (Var "f")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "R"))) "is" ($#v1_tops_2 :::"open"::: ) )))) ; theorem :: PSCOMP_1:13 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "R")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) & (Bool (Set (Var "R")) "is" ($#v2_measure6 :::"closed"::: ) )) "holds" (Bool (Set (Set "(" ($#k3_measure6 :::"""::: ) (Set (Var "f")) ")" ) ($#k7_relset_1 :::".:"::: ) (Set (Var "R"))) "is" ($#v2_tops_2 :::"closed"::: ) )))) ; definitionlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "f" be ($#m1_subset_1 :::"RealMap":::) "of" (Set (Const "T")); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); :: original: :::"|"::: redefine func "f" :::"|"::: "X" -> ($#m1_subset_1 :::"RealMap":::) "of" (Set "(" "T" ($#k1_pre_topc :::"|"::: ) "X" ")" ); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "f" be ($#v1_pscomp_1 :::"continuous"::: ) ($#m1_subset_1 :::"RealMap":::) "of" (Set (Const "T")); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "X") -> ($#v1_pscomp_1 :::"continuous"::: ) for ($#m1_subset_1 :::"RealMap":::) "of" (Set "(" "T" ($#k1_pre_topc :::"|"::: ) "X" ")" ); end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "P" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "T")); cluster (Set "T" ($#k1_pre_topc :::"|"::: ) "P") -> ($#v1_compts_1 :::"compact"::: ) ; end; begin theorem :: PSCOMP_1:14 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool "(" "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v5_measure6 :::"with_max"::: ) ) ")" ) "iff" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v6_measure6 :::"with_min"::: ) )) ")" )) ; theorem :: PSCOMP_1:15 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool "(" "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) "iff" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v5_measure6 :::"with_max"::: ) )) ")" )) ; definitionlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; attr "T" is :::"pseudocompact"::: means :: PSCOMP_1:def 4 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" "T" "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) )); end; :: deftheorem defines :::"pseudocompact"::: PSCOMP_1:def 4 : (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v2_pscomp_1 :::"pseudocompact"::: ) ) "iff" (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) )) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_compts_1 :::"compact"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pscomp_1 :::"pseudocompact"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pre_topc :::"TopSpace-like"::: ) ($#v1_compts_1 :::"compact"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registrationlet "T" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_pscomp_1 :::"pseudocompact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FUNCT_2((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set ($#k1_numbers :::"REAL"::: ) )) ($#v1_pscomp_1 :::"continuous"::: ) -> ($#v5_measure6 :::"with_max"::: ) ($#v6_measure6 :::"with_min"::: ) ($#v1_comseq_2 :::"bounded"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T") "," (Set ($#k1_numbers :::"REAL"::: ) ))))); end; theorem :: PSCOMP_1:16 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "f")) "being" ($#v1_pscomp_1 :::"continuous"::: ) ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set (Var "f")) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "Y")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set (Var "f")) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" ))))))) ; theorem :: PSCOMP_1:17 (Bool "for" (Set (Var "T")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "f")) "being" ($#v1_pscomp_1 :::"continuous"::: ) ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set (Var "f")) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set (Var "f")) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "Y")) ")" ))))))) ; begin registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "X", "Y" be ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); cluster (Set "X" ($#k3_xboole_0 :::"/\"::: ) "Y") -> ($#v2_compts_1 :::"compact"::: ) for ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) "n" ")" ); end; definitionfunc :::"proj1"::: -> ($#m1_subset_1 :::"RealMap":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: PSCOMP_1:def 5 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ))); func :::"proj2"::: -> ($#m1_subset_1 :::"RealMap":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) means :: PSCOMP_1:def 6 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ))); end; :: deftheorem defines :::"proj1"::: PSCOMP_1:def 5 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k4_pscomp_1 :::"proj1"::: ) )) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ))) ")" )); :: deftheorem defines :::"proj2"::: PSCOMP_1:def 6 : (Bool "for" (Set (Var "b1")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Var "b1")) ($#r1_hidden :::"="::: ) (Set ($#k5_pscomp_1 :::"proj2"::: ) )) "iff" (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set (Var "b1")) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ))) ")" )); theorem :: PSCOMP_1:18 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k8_relset_1 :::"""::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "r2")) ($#k19_euclid :::"]|"::: ) ) where r1, r2 "is" ($#m1_subset_1 :::"Real":::) : (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) ")" ) "}" )) ; theorem :: PSCOMP_1:19 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r3")) "," (Set (Var "q3")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "r2")) ($#k19_euclid :::"]|"::: ) ) where r1, r2 "is" ($#m1_subset_1 :::"Real":::) : (Bool "(" (Bool (Set (Var "r3")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r1"))) & (Bool (Set (Var "r1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q3"))) ")" ) "}" )) "holds" (Bool (Set (Var "P")) "is" ($#v3_pre_topc :::"open"::: ) ))) ; theorem :: PSCOMP_1:20 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k8_relset_1 :::"""::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k2_rcomp_1 :::".["::: ) )) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "r2")) ($#k19_euclid :::"]|"::: ) ) where r1, r2 "is" ($#m1_subset_1 :::"Real":::) : (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) ")" ) "}" )) ; theorem :: PSCOMP_1:21 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "r3")) "," (Set (Var "q3")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) "{" (Set ($#k19_euclid :::"|["::: ) (Set (Var "r1")) "," (Set (Var "r2")) ($#k19_euclid :::"]|"::: ) ) where r1, r2 "is" ($#m1_subset_1 :::"Real":::) : (Bool "(" (Bool (Set (Var "r3")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r2"))) & (Bool (Set (Var "r2")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q3"))) ")" ) "}" )) "holds" (Bool (Set (Var "P")) "is" ($#v3_pre_topc :::"open"::: ) ))) ; registration cluster (Set ($#k4_pscomp_1 :::"proj1"::: ) ) -> ($#v1_pscomp_1 :::"continuous"::: ) ; cluster (Set ($#k5_pscomp_1 :::"proj2"::: ) ) -> ($#v1_pscomp_1 :::"continuous"::: ) ; end; theorem :: PSCOMP_1:22 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )))) ; theorem :: PSCOMP_1:23 (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )))) ; definitionlet "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"W-bound"::: "X" -> ($#m1_subset_1 :::"Real":::) equals :: PSCOMP_1:def 7 (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) "X" ")" )); func :::"N-bound"::: "X" -> ($#m1_subset_1 :::"Real":::) equals :: PSCOMP_1:def 8 (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) "X" ")" )); func :::"E-bound"::: "X" -> ($#m1_subset_1 :::"Real":::) equals :: PSCOMP_1:def 9 (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) "X" ")" )); func :::"S-bound"::: "X" -> ($#m1_subset_1 :::"Real":::) equals :: PSCOMP_1:def 10 (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) "X" ")" )); end; :: deftheorem defines :::"W-bound"::: PSCOMP_1:def 7 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" )))); :: deftheorem defines :::"N-bound"::: PSCOMP_1:def 8 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" )))); :: deftheorem defines :::"E-bound"::: PSCOMP_1:def 9 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" )))); :: deftheorem defines :::"S-bound"::: PSCOMP_1:def 10 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set (Var "X")) ")" )))); theorem :: PSCOMP_1:24 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "(" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X")))) & (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X")))) ")" ))) ; definitionlet "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"SW-corner"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 11 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) "X" ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"NW-corner"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 12 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) "X" ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"NE-corner"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 13 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) "X" ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"SE-corner"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 14 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) "X" ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); end; :: deftheorem defines :::"SW-corner"::: PSCOMP_1:def 11 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"NW-corner"::: PSCOMP_1:def 12 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"NE-corner"::: PSCOMP_1:def 13 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"SE-corner"::: PSCOMP_1:def 14 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); theorem :: PSCOMP_1:25 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: PSCOMP_1:26 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ))) ; theorem :: PSCOMP_1:27 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ))) ; theorem :: PSCOMP_1:28 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ))) ; definitionlet "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"W-most"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 15 (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) "X" ")" ) "," (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) "X" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) "X"); func :::"N-most"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 16 (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) "X" ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) "X" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) "X"); func :::"E-most"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 17 (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) "X" ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) "X" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) "X"); func :::"S-most"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 18 (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) "X" ")" ) "," (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) "X" ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) "X"); end; :: deftheorem defines :::"W-most"::: PSCOMP_1:def 15 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))))); :: deftheorem defines :::"N-most"::: PSCOMP_1:def 16 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))))); :: deftheorem defines :::"E-most"::: PSCOMP_1:def 17 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))))); :: deftheorem defines :::"S-most"::: PSCOMP_1:def 18 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))))); registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); cluster (Set ($#k14_pscomp_1 :::"W-most"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ; cluster (Set ($#k15_pscomp_1 :::"N-most"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ; cluster (Set ($#k16_pscomp_1 :::"E-most"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ; cluster (Set ($#k17_pscomp_1 :::"S-most"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ; end; definitionlet "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ); func :::"W-min"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 19 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) "X" ")" ) "," (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) "X" ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"W-max"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 20 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) "X" ")" ) "," (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) "X" ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"N-min"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 21 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) "X" ")" ) ")" ) ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"N-max"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 22 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) "X" ")" ) ")" ) ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"E-max"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 23 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) "X" ")" ) "," (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) "X" ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"E-min"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 24 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) "X" ")" ) "," (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) "X" ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"S-max"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 25 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) "X" ")" ) ")" ) ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); func :::"S-min"::: "X" -> ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) equals :: PSCOMP_1:def 26 (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) "X" ")" ) ")" ) ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) "X" ")" ) ($#k19_euclid :::"]|"::: ) ); end; :: deftheorem defines :::"W-min"::: PSCOMP_1:def 19 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"W-max"::: PSCOMP_1:def 20 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"N-min"::: PSCOMP_1:def 21 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"N-max"::: PSCOMP_1:def 22 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) "," (Set "(" ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"E-max"::: PSCOMP_1:def 23 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"E-min"::: PSCOMP_1:def 24 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"S-max"::: PSCOMP_1:def 25 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k2_pscomp_1 :::"upper_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); :: deftheorem defines :::"S-min"::: PSCOMP_1:def 26 : (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_euclid :::"|["::: ) (Set "(" ($#k1_pscomp_1 :::"lower_bound"::: ) (Set "(" (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k3_pscomp_1 :::"|"::: ) (Set "(" ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X")) ")" ) ")" ) ")" ) "," (Set "(" ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X")) ")" ) ($#k19_euclid :::"]|"::: ) ))); theorem :: PSCOMP_1:29 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: PSCOMP_1:30 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )) ; theorem :: PSCOMP_1:31 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Z")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "Z"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "Z")) ")" ) ($#k17_euclid :::"`1"::: ) )) & "(" (Bool (Bool (Set (Var "Z")) "is" ($#v2_compts_1 :::"compact"::: ) )) "implies" (Bool "(" (Bool (Set (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "Z")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "Z")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ")" ))) ; theorem :: PSCOMP_1:32 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:33 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X")) ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:34 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X")))) & (Bool (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X")))) ")" )) ; theorem :: PSCOMP_1:35 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) ")" )) ; theorem :: PSCOMP_1:36 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "X"))))) "holds" (Bool (Set ($#k14_pscomp_1 :::"W-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: PSCOMP_1:37 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )) ; theorem :: PSCOMP_1:38 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: PSCOMP_1:39 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Z")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "Z"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "Z")) ")" ) ($#k18_euclid :::"`2"::: ) )) & "(" (Bool (Bool (Set (Var "Z")) "is" ($#v2_compts_1 :::"compact"::: ) )) "implies" (Bool "(" (Bool (Set (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "Z")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "Z")) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" ) ")" ")" ))) ; theorem :: PSCOMP_1:40 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:41 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X")) ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:42 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X")))) & (Bool (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X")))) ")" )) ; theorem :: PSCOMP_1:43 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) ")" )) ; theorem :: PSCOMP_1:44 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "X"))))) "holds" (Bool (Set ($#k15_pscomp_1 :::"N-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: PSCOMP_1:45 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "P")) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: PSCOMP_1:46 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )) ; theorem :: PSCOMP_1:47 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Z")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "Z"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "Z")) ")" ) ($#k17_euclid :::"`1"::: ) )) & "(" (Bool (Bool (Set (Var "Z")) "is" ($#v2_compts_1 :::"compact"::: ) )) "implies" (Bool "(" (Bool (Set (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "Z")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "Z")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" ) ")" ")" ))) ; theorem :: PSCOMP_1:48 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:49 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X")) ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:50 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X")))) & (Bool (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X")))) ")" )) ; theorem :: PSCOMP_1:51 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) ")" )) ; theorem :: PSCOMP_1:52 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "X"))))) "holds" (Bool (Set ($#k16_pscomp_1 :::"E-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: PSCOMP_1:53 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) & (Bool (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "P")) ")" ) ($#k18_euclid :::"`2"::: ) )) ")" )) ; theorem :: PSCOMP_1:54 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" )) ; theorem :: PSCOMP_1:55 (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Z")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "Z"))))) "holds" (Bool "(" (Bool (Set (Set (Var "p")) ($#k18_euclid :::"`2"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "Z")) ")" ) ($#k18_euclid :::"`2"::: ) )) & "(" (Bool (Bool (Set (Var "Z")) "is" ($#v2_compts_1 :::"compact"::: ) )) "implies" (Bool "(" (Bool (Set (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "Z")) ")" ) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) )) & (Bool (Set (Set (Var "p")) ($#k17_euclid :::"`1"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "Z")) ")" ) ($#k17_euclid :::"`1"::: ) )) ")" ) ")" ")" ))) ; theorem :: PSCOMP_1:56 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X"))) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:57 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X")) ")" ) ")" ) ($#r1_tarski :::"c="::: ) (Set ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: PSCOMP_1:58 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X")))) & (Bool (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X"))) ($#r2_hidden :::"in"::: ) (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X")))) ")" )) ; theorem :: PSCOMP_1:59 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set (Set "(" ($#k1_rltopsp1 :::"LSeg"::: ) "(" (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "X")) ")" ) ")" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) )) ")" )) ; theorem :: PSCOMP_1:60 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "X"))))) "holds" (Bool (Set ($#k17_pscomp_1 :::"S-most"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "X")) ")" ) ($#k1_tarski :::"}"::: ) ))) ; theorem :: PSCOMP_1:61 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k20_pscomp_1 :::"N-min"::: ) (Set (Var "P"))))) "holds" (Bool (Set ($#k19_pscomp_1 :::"W-max"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k11_pscomp_1 :::"NW-corner"::: ) (Set (Var "P"))))) ; theorem :: PSCOMP_1:62 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k22_pscomp_1 :::"E-max"::: ) (Set (Var "P"))))) "holds" (Bool (Set ($#k21_pscomp_1 :::"N-max"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k12_pscomp_1 :::"NE-corner"::: ) (Set (Var "P"))))) ; theorem :: PSCOMP_1:63 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k24_pscomp_1 :::"S-max"::: ) (Set (Var "P"))))) "holds" (Bool (Set ($#k23_pscomp_1 :::"E-min"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k13_pscomp_1 :::"SE-corner"::: ) (Set (Var "P"))))) ; theorem :: PSCOMP_1:64 (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k18_pscomp_1 :::"W-min"::: ) (Set (Var "P"))))) "holds" (Bool (Set ($#k25_pscomp_1 :::"S-min"::: ) (Set (Var "P"))) ($#r1_hidden :::"="::: ) (Set ($#k10_pscomp_1 :::"SW-corner"::: ) (Set (Var "P"))))) ; theorem :: PSCOMP_1:65 (Bool "for" (Set (Var "r")) "," (Set (Var "s")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Set ($#k5_pscomp_1 :::"proj2"::: ) ) ($#k1_seq_1 :::"."::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k19_euclid :::"]|"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "s"))) & (Bool (Set (Set ($#k4_pscomp_1 :::"proj1"::: ) ) ($#k1_seq_1 :::"."::: ) (Set ($#k19_euclid :::"|["::: ) (Set (Var "r")) "," (Set (Var "s")) ($#k19_euclid :::"]|"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "r"))) ")" )) ; theorem :: PSCOMP_1:66 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Y")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k7_pscomp_1 :::"N-bound"::: ) (Set (Var "Y")))))) ; theorem :: PSCOMP_1:67 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Y")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k8_pscomp_1 :::"E-bound"::: ) (Set (Var "Y")))))) ; theorem :: PSCOMP_1:68 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Y")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k9_pscomp_1 :::"S-bound"::: ) (Set (Var "Y")))))) ; theorem :: PSCOMP_1:69 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "Y")) "being" ($#v2_compts_1 :::"compact"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Num 2) ")" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set (Var "Y")))) "holds" (Bool (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "X"))) ($#r1_xxreal_0 :::">="::: ) (Set ($#k6_pscomp_1 :::"W-bound"::: ) (Set (Var "Y")))))) ;