:: CONNSP_1 semantic presentation begin definitionlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "A", "B" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "GX")); pred "A" "," "B" :::"are_separated"::: means :: CONNSP_1:def 1 (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) "A") ($#r1_xboole_0 :::"misses"::: ) "B") & (Bool "A" ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) "B")) ")" ); symmetry (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "GX")) "st" (Bool (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "B"))) & (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "B"))))) "holds" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "B"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "A"))) & (Bool (Set (Var "B")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A")))) ")" )) ; end; :: deftheorem defines :::"are_separated"::: CONNSP_1:def 1 : (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ) "iff" (Bool "(" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "B"))) & (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "B")))) ")" ) ")" ))); theorem :: CONNSP_1:1 (Bool "for" (Set (Var "TS")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "K")) "," (Set (Var "L")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "TS")) "st" (Bool (Bool (Set (Var "K")) "," (Set (Var "L")) ($#r1_connsp_1 :::"are_separated"::: ) )) "holds" (Bool (Set (Var "K")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "L"))))) ; theorem :: CONNSP_1:2 (Bool "for" (Set (Var "TS")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "K")) "," (Set (Var "L")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "TS")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "L")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "K")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "L")))) "holds" (Bool (Set (Var "K")) "," (Set (Var "L")) ($#r1_connsp_1 :::"are_separated"::: ) ))) ; theorem :: CONNSP_1:3 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) & (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ))) ; theorem :: CONNSP_1:4 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) & (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) )) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ))) ; theorem :: CONNSP_1:5 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X9")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "GX")) (Bool "for" (Set (Var "P1")) "," (Set (Var "Q1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X9")) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set (Var "P1"))) & (Bool (Set (Var "Q")) ($#r1_hidden :::"="::: ) (Set (Var "Q1"))) & (Bool (Set (Var "P")) "," (Set (Var "Q")) ($#r1_connsp_1 :::"are_separated"::: ) )) "holds" (Bool (Set (Var "P1")) "," (Set (Var "Q1")) ($#r1_connsp_1 :::"are_separated"::: ) ))))) ; theorem :: CONNSP_1:6 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X9")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "GX")) (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) (Bool "for" (Set (Var "P1")) "," (Set (Var "Q1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X9")) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set (Var "P1"))) & (Bool (Set (Var "Q")) ($#r1_hidden :::"="::: ) (Set (Var "Q1"))) & (Bool (Set (Set (Var "P")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Q"))) ($#r1_tarski :::"c="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "X9")))) & (Bool (Set (Var "P")) "," (Set (Var "Q")) ($#r1_connsp_1 :::"are_separated"::: ) )) "holds" (Bool (Set (Var "P1")) "," (Set (Var "Q1")) ($#r1_connsp_1 :::"are_separated"::: ) ))))) ; theorem :: CONNSP_1:7 (Bool "for" (Set (Var "TS")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "K")) "," (Set (Var "L")) "," (Set (Var "K1")) "," (Set (Var "L1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "TS")) "st" (Bool (Bool (Set (Var "K")) "," (Set (Var "L")) ($#r1_connsp_1 :::"are_separated"::: ) ) & (Bool (Set (Var "K1")) ($#r1_tarski :::"c="::: ) (Set (Var "K"))) & (Bool (Set (Var "L1")) ($#r1_tarski :::"c="::: ) (Set (Var "L")))) "holds" (Bool (Set (Var "K1")) "," (Set (Var "L1")) ($#r1_connsp_1 :::"are_separated"::: ) ))) ; theorem :: CONNSP_1:8 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ) & (Bool (Set (Var "A")) "," (Set (Var "C")) ($#r1_connsp_1 :::"are_separated"::: ) )) "holds" (Bool (Set (Var "A")) "," (Set (Set (Var "B")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C"))) ($#r1_connsp_1 :::"are_separated"::: ) ))) ; theorem :: CONNSP_1:9 (Bool "for" (Set (Var "TS")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "K")) "," (Set (Var "L")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "TS")) "st" (Bool (Bool "(" (Bool "(" (Bool (Set (Var "K")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "L")) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ) "or" (Bool "(" (Bool (Set (Var "K")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "L")) "is" ($#v3_pre_topc :::"open"::: ) ) ")" ) ")" )) "holds" (Bool (Set (Set (Var "K")) ($#k7_subset_1 :::"\"::: ) (Set (Var "L"))) "," (Set (Set (Var "L")) ($#k7_subset_1 :::"\"::: ) (Set (Var "K"))) ($#r1_connsp_1 :::"are_separated"::: ) ))) ; definitionlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; attr "GX" is :::"connected"::: means :: CONNSP_1:def 2 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" "GX" "st" (Bool (Bool (Set ($#k2_struct_0 :::"[#]"::: ) "GX") ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) & (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ) & (Bool (Bool "not" (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) "GX")))) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) "GX"))); end; :: deftheorem defines :::"connected"::: CONNSP_1:def 2 : (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) "iff" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) & (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ) & (Bool (Bool "not" (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))))) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX"))))) ")" )); theorem :: CONNSP_1:10 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) "iff" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))) & (Bool (Set (Var "B")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "B")))) ")" )) ; theorem :: CONNSP_1:11 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) "iff" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))) & (Bool (Set (Var "B")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))) & (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_pre_topc :::"open"::: ) )) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "B")))) ")" )) ; theorem :: CONNSP_1:12 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set "(" (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set (Var "A")) ")" )))) ")" )) ; theorem :: CONNSP_1:13 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Bool "not" (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))))) ")" )) ; theorem :: CONNSP_1:14 (Bool "for" (Set (Var "GX")) "," (Set (Var "GY")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "GX")) "," (Set (Var "GY")) "st" (Bool (Bool (Set (Var "F")) "is" ($#v5_pre_topc :::"continuous"::: ) ) & (Bool (Set (Set (Var "F")) ($#k7_relset_1 :::".:"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GY")))) & (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) )) "holds" (Bool (Set (Var "GY")) "is" ($#v1_connsp_1 :::"connected"::: ) ))) ; definitionlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "GX")); attr "A" is :::"connected"::: means :: CONNSP_1:def 3 (Bool (Set "GX" ($#k1_pre_topc :::"|"::: ) "A") "is" ($#v1_connsp_1 :::"connected"::: ) ); end; :: deftheorem defines :::"connected"::: CONNSP_1:def 3 : (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) "iff" (Bool (Set (Set (Var "GX")) ($#k1_pre_topc :::"|"::: ) (Set (Var "A"))) "is" ($#v1_connsp_1 :::"connected"::: ) ) ")" ))); theorem :: CONNSP_1:15 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) "iff" (Bool "for" (Set (Var "P")) "," (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set (Var "P")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "Q")))) & (Bool (Set (Var "P")) "," (Set (Var "Q")) ($#r1_connsp_1 :::"are_separated"::: ) ) & (Bool (Bool "not" (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))))) "holds" (Bool (Set (Var "Q")) ($#r1_hidden :::"="::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX"))))) ")" ))) ; theorem :: CONNSP_1:16 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C")))) & (Bool (Set (Var "B")) "," (Set (Var "C")) ($#r1_connsp_1 :::"are_separated"::: ) ) & (Bool (Bool "not" (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))))) "holds" (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "C"))))) ; theorem :: CONNSP_1:17 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "B")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Bool "not" (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ))) "holds" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v2_connsp_1 :::"connected"::: ) ))) ; theorem :: CONNSP_1:18 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "C")) "," (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "A"))) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "C"))))) "holds" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ))) ; theorem :: CONNSP_1:19 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) )) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) "is" ($#v2_connsp_1 :::"connected"::: ) ))) ; theorem :: CONNSP_1:20 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C")))) & (Bool (Set (Var "B")) "," (Set (Var "C")) ($#r1_connsp_1 :::"are_separated"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C"))) "is" ($#v2_connsp_1 :::"connected"::: ) ) ")" ))) ; theorem :: CONNSP_1:21 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C")))) & (Bool (Set (Var "B")) "," (Set (Var "C")) ($#r1_connsp_1 :::"are_separated"::: ) ) & (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "C"))) "is" ($#v4_pre_topc :::"closed"::: ) ) ")" ))) ; theorem :: CONNSP_1:22 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "C")) "," (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "C")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "A"))) & (Bool (Set (Set (Var "C")) ($#k7_subset_1 :::"\"::: ) (Set (Var "A"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX"))))) "holds" (Bool (Set (Var "C")) ($#r1_xboole_0 :::"meets"::: ) (Set ($#k2_tops_1 :::"Fr"::: ) (Set (Var "A")))))) ; theorem :: CONNSP_1:23 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X9")) "being" ($#m1_pre_topc :::"SubSpace"::: ) "of" (Set (Var "GX")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) (Bool "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X9")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B")))) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) "iff" (Bool (Set (Var "B")) "is" ($#v2_connsp_1 :::"connected"::: ) ) ")" ))))) ; theorem :: CONNSP_1:24 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Var "B")) "is" ($#v4_pre_topc :::"closed"::: ) ) & (Bool (Set (Set (Var "A")) ($#k4_subset_1 :::"\/"::: ) (Set (Var "B"))) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Set (Var "A")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "B"))) "is" ($#v2_connsp_1 :::"connected"::: ) )) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "B")) "is" ($#v2_connsp_1 :::"connected"::: ) ) ")" ))) ; theorem :: CONNSP_1:25 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "GX")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) ")" ) & (Bool "ex" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool "(" (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))) & (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "B")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "B")) ($#r1_hidden :::"<>"::: ) (Set (Var "A")))) "holds" (Bool "not" (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) )) ")" ) ")" ))) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))) "is" ($#v2_connsp_1 :::"connected"::: ) ))) ; theorem :: CONNSP_1:26 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "GX")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) ")" ) & (Bool (Set ($#k6_setfam_1 :::"meet"::: ) (Set (Var "F"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX"))))) "holds" (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))) "is" ($#v2_connsp_1 :::"connected"::: ) ))) ; theorem :: CONNSP_1:27 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX"))) "is" ($#v2_connsp_1 :::"connected"::: ) ) "iff" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) ")" )) ; registrationlet "GX" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "GX")); cluster (Set ($#k1_tarski :::"{"::: ) "x" ($#k1_tarski :::"}"::: ) ) -> ($#v2_connsp_1 :::"connected"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "GX"; end; definitionlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "GX")); pred "x" "," "y" :::"are_joined"::: means :: CONNSP_1:def 4 (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" "GX" "st" (Bool "(" (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool "x" ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool "y" ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )); end; :: deftheorem defines :::"are_joined"::: CONNSP_1:def 4 : (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r2_connsp_1 :::"are_joined"::: ) ) "iff" (Bool "ex" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool "(" (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) ")" ))); theorem :: CONNSP_1:28 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "st" (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r2_connsp_1 :::"are_joined"::: ) )))) "holds" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) )) ; theorem :: CONNSP_1:29 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool "(" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "st" (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r2_connsp_1 :::"are_joined"::: ) ))) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r2_connsp_1 :::"are_joined"::: ) )) ")" )) ; theorem :: CONNSP_1:30 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "st" (Bool (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r2_connsp_1 :::"are_joined"::: ) ) ")" )) "holds" (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) )) ; theorem :: CONNSP_1:31 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "GX")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "iff" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" ) ")" )) "holds" (Bool (Set (Var "F")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ))))) ; definitionlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "A" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "GX")); attr "A" is :::"a_component"::: means :: CONNSP_1:def 5 (Bool "(" (Bool "A" "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" "GX" "st" (Bool (Bool (Set (Var "B")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool "A" ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool "A" ($#r1_hidden :::"="::: ) (Set (Var "B"))) ")" ) ")" ); end; :: deftheorem defines :::"a_component"::: CONNSP_1:def 5 : (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) "iff" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool "(" "for" (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "B")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B"))) ")" ) ")" ) ")" ))); registrationlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v3_connsp_1 :::"a_component"::: ) -> ($#v2_connsp_1 :::"connected"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "GX"))); end; registrationlet "GX" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster ($#v3_connsp_1 :::"a_component"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "GX"))); end; theorem :: CONNSP_1:32 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) )) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_struct_0 :::"{}"::: ) (Set (Var "GX")))))) ; registrationlet "GX" be ($#l1_pre_topc :::"TopSpace":::); cluster ($#v3_connsp_1 :::"a_component"::: ) -> ($#v4_pre_topc :::"closed"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "GX"))); end; theorem :: CONNSP_1:33 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) )) "holds" (Bool (Set (Var "A")) "is" ($#v4_pre_topc :::"closed"::: ) ))) ; theorem :: CONNSP_1:34 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) & (Bool (Bool "not" (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B"))))) "holds" (Bool (Set (Var "A")) "," (Set (Var "B")) ($#r1_connsp_1 :::"are_separated"::: ) ))) ; theorem :: CONNSP_1:35 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) & (Bool (Set (Var "B")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) & (Bool (Bool "not" (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Var "B"))))) "holds" (Bool (Set (Var "A")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "B"))))) ; theorem :: CONNSP_1:36 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) )) "holds" (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" "not" (Bool (Set (Var "S")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) "or" (Bool (Set (Var "C")) ($#r1_xboole_0 :::"misses"::: ) (Set (Var "S"))) "or" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "S"))) ")" )))) ; definitionlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "A", "B" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "GX")); pred "B" :::"is_a_component_of"::: "A" means :: CONNSP_1:def 6 (Bool "ex" (Set (Var "B1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "GX" ($#k1_pre_topc :::"|"::: ) "A" ")" ) "st" (Bool "(" (Bool (Set (Var "B1")) ($#r1_hidden :::"="::: ) "B") & (Bool (Set (Var "B1")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) ")" )); end; :: deftheorem defines :::"is_a_component_of"::: CONNSP_1:def 6 : (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "B")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Var "A"))) "iff" (Bool "ex" (Set (Var "B1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "GX")) ($#k1_pre_topc :::"|"::: ) (Set (Var "A")) ")" ) "st" (Bool "(" (Bool (Set (Var "B1")) ($#r1_hidden :::"="::: ) (Set (Var "B"))) & (Bool (Set (Var "B1")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) ")" )) ")" ))); theorem :: CONNSP_1:37 (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "," (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "GX")) "is" ($#v1_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "C")) ($#r3_connsp_1 :::"is_a_component_of"::: ) (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "GX")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set (Var "C"))) "is" ($#v2_connsp_1 :::"connected"::: ) ))) ; definitionlet "GX" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "GX")); func :::"Component_of"::: "x" -> ($#m1_subset_1 :::"Subset":::) "of" "GX" means :: CONNSP_1:def 7 (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" "GX" "st" (Bool "(" (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "GX" "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "iff" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool "x" ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" ) ")" ) & (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) it) ")" )); end; :: deftheorem defines :::"Component_of"::: CONNSP_1:def 7 : (Bool "for" (Set (Var "GX")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x")))) "iff" (Bool "ex" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "GX")) "st" (Bool "(" (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "iff" (Bool "(" (Bool (Set (Var "A")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" ) ")" ) & (Bool (Set ($#k5_setfam_1 :::"union"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Var "b3"))) ")" )) ")" )))); theorem :: CONNSP_1:38 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x")))))) ; registrationlet "GX" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); let "x" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "GX")); cluster (Set ($#k1_connsp_1 :::"Component_of"::: ) "x") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_connsp_1 :::"connected"::: ) ; end; theorem :: CONNSP_1:39 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "C")) "is" ($#v2_connsp_1 :::"connected"::: ) ) & (Bool (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x"))) ($#r1_tarski :::"c="::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x"))))))) ; theorem :: CONNSP_1:40 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) "iff" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x"))))) ")" ))) ; theorem :: CONNSP_1:41 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "A")))) "holds" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x"))))))) ; theorem :: CONNSP_1:42 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "," (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "GX")) "st" (Bool (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x"))))) "holds" (Bool (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k1_connsp_1 :::"Component_of"::: ) (Set (Var "x")))))) ; theorem :: CONNSP_1:43 (Bool "for" (Set (Var "GX")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set (Var "GX")) "st" (Bool (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "GX")) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) "iff" (Bool (Set (Var "A")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) ")" ) ")" )) "holds" (Bool (Set (Var "F")) "is" ($#m1_setfam_1 :::"Cover":::) "of" (Set (Var "GX"))))) ; begin registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v1_xboole_0 :::"empty"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T"))); end; registrationlet "T" be ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#v1_xboole_0 :::"empty"::: ) -> ($#v2_connsp_1 :::"connected"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "T"))); end; theorem :: CONNSP_1:44 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v2_connsp_1 :::"connected"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T"))) "iff" (Bool (Set (Var "X")) "is" ($#v2_connsp_1 :::"connected"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" )) ")" ))) ; theorem :: CONNSP_1:45 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "T"))) "," (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "T"))) "#)" ) "st" (Bool (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "Y")))) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) "iff" (Bool (Set (Var "Y")) "is" ($#v3_connsp_1 :::"a_component"::: ) ) ")" )))) ; theorem :: CONNSP_1:46 (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "P")) "," (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "Q")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "G")) ($#k1_pre_topc :::"|"::: ) (Set (Var "A")) ")" ) "st" (Bool (Bool (Set (Var "P")) ($#r1_hidden :::"="::: ) (Set (Var "Q"))) & (Bool (Set (Var "P")) "is" ($#v2_connsp_1 :::"connected"::: ) )) "holds" (Bool (Set (Var "Q")) "is" ($#v2_connsp_1 :::"connected"::: ) )))) ;