:: PENCIL_1 semantic presentation begin theorem :: PENCIL_1:1 (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set ($#k4_card_3 :::"product"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "g")))) & (Bool (Set (Var "f")) "is" ($#v2_relat_1 :::"non-empty"::: ) )) "holds" (Bool (Set (Var "g")) "is" ($#v2_relat_1 :::"non-empty"::: ) )) ; theorem :: PENCIL_1:2 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X")))) "iff" (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" )) ")" )) ; theorem :: PENCIL_1:3 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) ")" )))) ; theorem :: PENCIL_1:4 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X")))) "iff" (Bool (Bool "not" (Set (Var "X")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) ")" )) ; theorem :: PENCIL_1:5 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Num 3) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X")))) "iff" (Bool "ex" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "y"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) ")" )) ")" )) ; theorem :: PENCIL_1:6 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Num 3) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))))) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) (Bool "ex" (Set (Var "z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set (Var "z"))) ")" )))) ; begin definitionlet "S" be ($#l1_pre_topc :::"TopStruct"::: ) ; mode Block of "S" is ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "S"); end; definitionlet "S" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "S")); pred "x" "," "y" :::"are_collinear"::: means :: PENCIL_1:def 1 (Bool "(" (Bool "x" ($#r1_hidden :::"="::: ) "y") "or" (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" "S" "st" (Bool (Set ($#k2_tarski :::"{"::: ) "x" "," "y" ($#k2_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "l")))) ")" ); end; :: deftheorem defines :::"are_collinear"::: PENCIL_1:def 1 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_pencil_1 :::"are_collinear"::: ) ) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y"))) "or" (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) "st" (Bool (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "l")))) ")" ) ")" ))); definitionlet "S" be ($#l1_pre_topc :::"TopStruct"::: ) ; let "T" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "S")); attr "T" is :::"closed_under_lines"::: means :: PENCIL_1:def 2 (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" "S" "st" (Bool (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "l")) ($#k9_subset_1 :::"/\"::: ) "T" ")" )))) "holds" (Bool (Set (Var "l")) ($#r1_tarski :::"c="::: ) "T")); attr "T" is :::"strong"::: means :: PENCIL_1:def 3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "T") & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "T")) "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_pencil_1 :::"are_collinear"::: ) )); end; :: deftheorem defines :::"closed_under_lines"::: PENCIL_1:def 2 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) "iff" (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) "st" (Bool (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "l")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "T")) ")" )))) "holds" (Bool (Set (Var "l")) ($#r1_tarski :::"c="::: ) (Set (Var "T")))) ")" ))); :: deftheorem defines :::"strong"::: PENCIL_1:def 3 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "T")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "T")) "is" ($#v2_pencil_1 :::"strong"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "T"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "T")))) "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_pencil_1 :::"are_collinear"::: ) )) ")" ))); definitionlet "S" be ($#l1_pre_topc :::"TopStruct"::: ) ; attr "S" is :::"void"::: means :: PENCIL_1:def 4 (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "S") "is" ($#v1_xboole_0 :::"empty"::: ) ); attr "S" is :::"degenerated"::: means :: PENCIL_1:def 5 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "is" ($#m1_subset_1 :::"Block":::) "of" "S"); attr "S" is :::"with_non_trivial_blocks"::: means :: PENCIL_1:def 6 (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Block":::) "of" "S" "holds" (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "k"))))); attr "S" is :::"identifying_close_blocks"::: means :: PENCIL_1:def 7 (Bool "for" (Set (Var "k")) "," (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" "S" "st" (Bool (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "k")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "l")) ")" )))) "holds" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set (Var "l")))); attr "S" is :::"truly-partial"::: means :: PENCIL_1:def 8 (Bool "not" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_pencil_1 :::"are_collinear"::: ) ))); attr "S" is :::"without_isolated_points"::: means :: PENCIL_1:def 9 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" "S" "st" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "l"))))); attr "S" is :::"connected"::: means :: PENCIL_1:def 10 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Var "a")) "," (Set (Var "b")) ($#r1_pencil_1 :::"are_collinear"::: ) )) ")" ) ")" ))); attr "S" is :::"strongly_connected"::: means :: PENCIL_1:def 11 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" "S" (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" "S" "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) & (Bool (Set (Var "X")) "is" ($#v2_pencil_1 :::"strong"::: ) )) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" "S" "st" (Bool (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "W")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) & (Bool (Set (Var "W")) "is" ($#v2_pencil_1 :::"strong"::: ) ) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))) ")" ) ")" )))); end; :: deftheorem defines :::"void"::: PENCIL_1:def 4 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v3_pencil_1 :::"void"::: ) ) "iff" (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) "is" ($#v1_xboole_0 :::"empty"::: ) ) ")" )); :: deftheorem defines :::"degenerated"::: PENCIL_1:def 5 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v4_pencil_1 :::"degenerated"::: ) ) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "is" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S"))) ")" )); :: deftheorem defines :::"with_non_trivial_blocks"::: PENCIL_1:def 6 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ) "iff" (Bool "for" (Set (Var "k")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) "holds" (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "k"))))) ")" )); :: deftheorem defines :::"identifying_close_blocks"::: PENCIL_1:def 7 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ) "iff" (Bool "for" (Set (Var "k")) "," (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) "st" (Bool (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set (Var "k")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "l")) ")" )))) "holds" (Bool (Set (Var "k")) ($#r1_hidden :::"="::: ) (Set (Var "l")))) ")" )); :: deftheorem defines :::"truly-partial"::: PENCIL_1:def 8 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v7_pencil_1 :::"truly-partial"::: ) ) "iff" (Bool "not" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_pencil_1 :::"are_collinear"::: ) ))) ")" )); :: deftheorem defines :::"without_isolated_points"::: PENCIL_1:def 9 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v8_pencil_1 :::"without_isolated_points"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) "st" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "l"))))) ")" )); :: deftheorem defines :::"connected"::: PENCIL_1:def 10 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v9_pencil_1 :::"connected"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) & (Bool (Set (Var "b")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Var "a")) "," (Set (Var "b")) ($#r1_pencil_1 :::"are_collinear"::: ) )) ")" ) ")" ))) ")" )); :: deftheorem defines :::"strongly_connected"::: PENCIL_1:def 11 : (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v10_pencil_1 :::"strongly_connected"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) & (Bool (Set (Var "X")) "is" ($#v2_pencil_1 :::"strong"::: ) )) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) "st" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool "(" "for" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "W")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool "(" (Bool (Set (Var "W")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) & (Bool (Set (Var "W")) "is" ($#v2_pencil_1 :::"strong"::: ) ) ")" ) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))))) "holds" (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k1_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ))) ")" ) ")" )))) ")" )); theorem :: PENCIL_1:7 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Num 3) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))))) "holds" (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "L")) where L "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) : (Bool (Num 2) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "L")))) "}" )) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "S")) "is" ($#v2_struct_0 :::"empty"::: ) )) & (Bool (Bool "not" (Set (Var "S")) "is" ($#v3_pencil_1 :::"void"::: ) )) & (Bool (Bool "not" (Set (Var "S")) "is" ($#v4_pencil_1 :::"degenerated"::: ) )) & (Bool (Bool "not" (Set (Var "S")) "is" ($#v7_pencil_1 :::"truly-partial"::: ) )) & (Bool (Set (Var "S")) "is" ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ) & (Bool (Set (Var "S")) "is" ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ) & (Bool (Set (Var "S")) "is" ($#v8_pencil_1 :::"without_isolated_points"::: ) ) ")" ))) ; theorem :: PENCIL_1:8 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Num 3) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))))) "holds" (Bool "for" (Set (Var "K")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set ($#k1_card_1 :::"card"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Num 2))) "holds" (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set "{" (Set (Var "L")) where L "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) : (Bool (Num 2) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "L")))) "}" ($#k6_subset_1 :::"\"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "K")) ($#k6_domain_1 :::"}"::: ) )))) "holds" (Bool "(" (Bool (Bool "not" (Set (Var "S")) "is" ($#v2_struct_0 :::"empty"::: ) )) & (Bool (Bool "not" (Set (Var "S")) "is" ($#v3_pencil_1 :::"void"::: ) )) & (Bool (Bool "not" (Set (Var "S")) "is" ($#v4_pencil_1 :::"degenerated"::: ) )) & (Bool (Set (Var "S")) "is" ($#v7_pencil_1 :::"truly-partial"::: ) ) & (Bool (Set (Var "S")) "is" ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ) & (Bool (Set (Var "S")) "is" ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ) & (Bool (Set (Var "S")) "is" ($#v8_pencil_1 :::"without_isolated_points"::: ) ) ")" )))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#~v4_pencil_1 "non" ($#v4_pencil_1 :::"degenerated"::: ) ) ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ($#~v7_pencil_1 "non" ($#v7_pencil_1 :::"truly-partial"::: ) ) ($#v8_pencil_1 :::"without_isolated_points"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_pre_topc :::"strict"::: ) ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#~v4_pencil_1 "non" ($#v4_pencil_1 :::"degenerated"::: ) ) ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ($#v7_pencil_1 :::"truly-partial"::: ) ($#v8_pencil_1 :::"without_isolated_points"::: ) for ($#l1_pre_topc :::"TopStruct"::: ) ; end; registrationlet "S" be ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) ; cluster (Set "the" ($#u1_pre_topc :::"topology"::: ) "of" "S") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "S" be ($#v8_pencil_1 :::"without_isolated_points"::: ) ($#l1_pre_topc :::"TopStruct"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Point":::) "of" (Set (Const "S")); redefine pred "x" "," "y" :::"are_collinear"::: means :: PENCIL_1:def 12 (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" "S" "st" (Bool (Set ($#k2_tarski :::"{"::: ) "x" "," "y" ($#k2_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "l")))); end; :: deftheorem defines :::"are_collinear"::: PENCIL_1:def 12 : (Bool "for" (Set (Var "S")) "being" ($#v8_pencil_1 :::"without_isolated_points"::: ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "x")) "," (Set (Var "y")) ($#r1_pencil_1 :::"are_collinear"::: ) ) "iff" (Bool "ex" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) "st" (Bool (Set ($#k2_tarski :::"{"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k2_tarski :::"}"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Var "l")))) ")" ))); definitionmode PLS is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#~v4_pencil_1 "non" ($#v4_pencil_1 :::"degenerated"::: ) ) ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ($#l1_pre_topc :::"TopStruct"::: ) ; end; definitionlet "F" be ($#m1_hidden :::"Relation":::); attr "F" is :::"TopStruct-yielding"::: means :: PENCIL_1:def 13 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "F"))) "holds" (Bool (Set (Var "x")) "is" ($#l1_pre_topc :::"TopStruct"::: ) )); end; :: deftheorem defines :::"TopStruct-yielding"::: PENCIL_1:def 13 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "x")) "is" ($#l1_pre_topc :::"TopStruct"::: ) )) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) -> ($#v2_pralg_1 :::"1-sorted-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#v11_pencil_1 :::"TopStruct-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "F" be ($#m1_hidden :::"Relation":::); attr "F" is :::"non-void-yielding"::: means :: PENCIL_1:def 14 (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "F"))) "holds" (Bool "not" (Bool (Set (Var "S")) "is" ($#v3_pencil_1 :::"void"::: ) ))); end; :: deftheorem defines :::"non-void-yielding"::: PENCIL_1:def 14 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v12_pencil_1 :::"non-void-yielding"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool "not" (Bool (Set (Var "S")) "is" ($#v3_pencil_1 :::"void"::: ) ))) ")" )); definitionlet "F" be ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"Function":::); redefine attr "F" is :::"non-void-yielding"::: means :: PENCIL_1:def 15 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "F"))) "holds" (Bool (Set (Var "i")) "is" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) )); end; :: deftheorem defines :::"non-void-yielding"::: PENCIL_1:def 15 : (Bool "for" (Set (Var "F")) "being" ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v12_pencil_1 :::"non-void-yielding"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "i")) "is" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) )) ")" )); definitionlet "F" be ($#m1_hidden :::"Relation":::); attr "F" is :::"trivial-yielding"::: means :: PENCIL_1:def 16 (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "F"))) "holds" (Bool (Set (Var "S")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )); end; :: deftheorem defines :::"trivial-yielding"::: PENCIL_1:def 16 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "S")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) ")" )); definitionlet "F" be ($#m1_hidden :::"Relation":::); attr "F" is :::"non-Trivial-yielding"::: means :: PENCIL_1:def 17 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "F"))) "holds" (Bool "not" (Bool (Set (Var "S")) "is" ($#v7_struct_0 :::"trivial"::: ) ))); end; :: deftheorem defines :::"non-Trivial-yielding"::: PENCIL_1:def 17 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v14_pencil_1 :::"non-Trivial-yielding"::: ) ) "iff" (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) "st" (Bool (Bool (Set (Var "S")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool "not" (Bool (Set (Var "S")) "is" ($#v7_struct_0 :::"trivial"::: ) ))) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v14_pencil_1 :::"non-Trivial-yielding"::: ) -> ($#v4_waybel_3 :::"non-Empty"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "F" be ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#m1_hidden :::"Function":::); redefine attr "F" is :::"non-Trivial-yielding"::: means :: PENCIL_1:def 18 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "F"))) "holds" (Bool (Set (Var "i")) "is" ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) )); end; :: deftheorem defines :::"non-Trivial-yielding"::: PENCIL_1:def 18 : (Bool "for" (Set (Var "F")) "being" ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#m1_hidden :::"Function":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v14_pencil_1 :::"non-Trivial-yielding"::: ) ) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "i")) "is" ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) )) ")" )); definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "j" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); :: original: :::"."::: redefine func "A" :::"."::: "j" -> ($#l1_pre_topc :::"TopStruct"::: ) ; end; definitionlet "F" be ($#m1_hidden :::"Relation":::); attr "F" is :::"PLS-yielding"::: means :: PENCIL_1:def 19 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "F"))) "holds" (Bool (Set (Var "x")) "is" ($#l1_pre_topc :::"PLS":::))); end; :: deftheorem defines :::"PLS-yielding"::: PENCIL_1:def 19 : (Bool "for" (Set (Var "F")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v15_pencil_1 :::"PLS-yielding"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Var "x")) "is" ($#l1_pre_topc :::"PLS":::))) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v15_pencil_1 :::"PLS-yielding"::: ) -> ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#v15_pencil_1 :::"PLS-yielding"::: ) -> ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#v12_pencil_1 :::"non-void-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#v15_pencil_1 :::"PLS-yielding"::: ) -> ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#v14_pencil_1 :::"non-Trivial-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "I" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#v15_pencil_1 :::"PLS-yielding"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v15_pencil_1 :::"PLS-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); let "j" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "I")); :: original: :::"."::: redefine func "A" :::"."::: "j" -> ($#l1_pre_topc :::"PLS":::); end; definitionlet "I" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); attr "A" is :::"Segre-like"::: means :: PENCIL_1:def 20 (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" "st" (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" "st" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool (Set "A" ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) "is" (Num 1) ($#v3_card_1 :::"-element"::: ) ))); end; :: deftheorem defines :::"Segre-like"::: PENCIL_1:def 20 : (Bool "for" (Set (Var "I")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "holds" (Bool "(" (Bool (Set (Var "A")) "is" ($#v16_pencil_1 :::"Segre-like"::: ) ) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "st" (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "st" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set (Var "j")))) "holds" (Bool (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) "is" (Num 1) ($#v3_card_1 :::"-element"::: ) ))) ")" ))); registrationlet "I" be ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k1_pzfmisc1 :::"{"::: ) "A" ($#k1_pzfmisc1 :::"}"::: ) ) -> ($#v13_pencil_1 :::"trivial-yielding"::: ) ; end; theorem :: PENCIL_1:9 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "for" (Set (Var "S")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Bool "not" (Set (Set (Var "A")) ($#k2_funct_7 :::"+*"::: ) "(" (Set (Var "i")) "," (Set (Var "S")) ")" ) "is" ($#v13_pencil_1 :::"trivial-yielding"::: ) )))))) ; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k1_pzfmisc1 :::"{"::: ) "A" ($#k1_pzfmisc1 :::"}"::: ) ) -> ($#v16_pencil_1 :::"Segre-like"::: ) ; end; theorem :: PENCIL_1:10 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "i")) "," (Set (Var "S")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k1_pzfmisc1 :::"{"::: ) (Set (Var "A")) ($#k1_pzfmisc1 :::"}"::: ) ) ($#k2_funct_7 :::"+*"::: ) "(" (Set (Var "i")) "," (Set (Var "S")) ")" ) "is" ($#v16_pencil_1 :::"Segre-like"::: ) )))) ; theorem :: PENCIL_1:11 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "B")) "being" ($#m1_pboole :::"Element"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "A"))) "holds" (Bool (Set ($#k1_pzfmisc1 :::"{"::: ) (Set (Var "B")) ($#k1_pzfmisc1 :::"}"::: ) ) "is" ($#m3_pboole :::"ManySortedSubset"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "A"))))))) ; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#v4_waybel_3 :::"non-Empty"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster ($#v1_relat_1 :::"Relation-like"::: ) bbbadV2_RELAT_1() "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#v13_pencil_1 :::"trivial-yielding"::: ) ($#v16_pencil_1 :::"Segre-like"::: ) for ($#m3_pboole :::"ManySortedSubset"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) "A"); end; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v2_pralg_1 :::"1-sorted-yielding"::: ) ($#v14_pencil_1 :::"non-Trivial-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster ($#v1_relat_1 :::"Relation-like"::: ) bbbadV2_RELAT_1() "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) for ($#m3_pboole :::"ManySortedSubset"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) "A"); end; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "B" be ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); func :::"indx"::: "B" -> ($#m1_subset_1 :::"Element"::: ) "of" "I" means :: PENCIL_1:def 21 (Bool (Bool "not" (Set "B" ($#k1_funct_1 :::"."::: ) it) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )); end; :: deftheorem defines :::"indx"::: PENCIL_1:def 21 : (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "B")) "being" ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_pencil_1 :::"indx"::: ) (Set (Var "B")))) "iff" (Bool (Bool "not" (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set (Var "b3"))) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) ")" )))); theorem :: PENCIL_1:12 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "st" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_pencil_1 :::"indx"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" (Num 1) ($#v3_card_1 :::"-element"::: ) )))) ; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "I" ($#v4_relat_1 :::"-defined"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_PARTFUN1("I") ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) -> bbbadV2_RELAT_1() for ($#m1_hidden :::"set"::: ) ; end; theorem :: PENCIL_1:13 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "holds" (Bool "(" (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k4_card_3 :::"product"::: ) (Set (Var "A")) ")" ))) "iff" (Bool "(" (Bool (Set (Var "A")) "is" bbbadV2_RELAT_1()) & (Bool (Bool "not" (Set (Var "A")) "is" ($#v13_pencil_1 :::"trivial-yielding"::: ) )) ")" ) ")" ))) ; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "B" be ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k4_card_3 :::"product"::: ) "B") -> ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ; end; begin definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); func :::"Segre_Blocks"::: "A" -> ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k12_pralg_1 :::"Carrier"::: ) "A" ")" ) ")" ) means :: PENCIL_1:def 22 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "B")) "being" ($#v16_pencil_1 :::"Segre-like"::: ) ($#m3_pboole :::"ManySortedSubset"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) "A") "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "B")))) & (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "I" "st" (Bool (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" ($#m1_subset_1 :::"Block":::) "of" (Set "(" "A" ($#k1_pencil_1 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"Segre_Blocks"::: PENCIL_1:def 22 : (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "A")) ")" ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_pencil_1 :::"Segre_Blocks"::: ) (Set (Var "A")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "B")) "being" ($#v16_pencil_1 :::"Segre-like"::: ) ($#m3_pboole :::"ManySortedSubset"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "A"))) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "B")))) & (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "st" (Bool (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) "is" ($#m1_subset_1 :::"Block":::) "of" (Set "(" (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i")) ")" ))) ")" )) ")" )) ")" )))); definitionlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); func :::"Segre_Product"::: "A" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) equals :: PENCIL_1:def 23 (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k12_pralg_1 :::"Carrier"::: ) "A" ")" ) ")" ) "," (Set "(" ($#k4_pencil_1 :::"Segre_Blocks"::: ) "A" ")" ) "#)" ); end; :: deftheorem defines :::"Segre_Product"::: PENCIL_1:def 23 : (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "holds" (Bool (Set ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#g1_pre_topc :::"TopStruct"::: ) "(#" (Set "(" ($#k4_card_3 :::"product"::: ) (Set "(" ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "A")) ")" ) ")" ) "," (Set "(" ($#k4_pencil_1 :::"Segre_Blocks"::: ) (Set (Var "A")) ")" ) "#)" )))); theorem :: PENCIL_1:14 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A")) ")" ) "holds" (Bool (Set (Var "x")) "is" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")))))) ; theorem :: PENCIL_1:15 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "not" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v3_pencil_1 :::"void"::: ) )))) "holds" (Bool "not" (Bool (Set ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A"))) "is" ($#v3_pencil_1 :::"void"::: ) )))) ; theorem :: PENCIL_1:16 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool "(" (Bool (Bool "not" (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v4_pencil_1 :::"degenerated"::: ) )) & (Bool "not" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v3_pencil_1 :::"void"::: ) ))) ")" ) ")" )) "holds" (Bool "not" (Bool (Set ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A"))) "is" ($#v4_pencil_1 :::"degenerated"::: ) )))) ; theorem :: PENCIL_1:17 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ) & (Bool "not" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v3_pencil_1 :::"void"::: ) ))) ")" ) ")" )) "holds" (Bool (Set ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A"))) "is" ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ))) ; theorem :: PENCIL_1:18 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v4_waybel_3 :::"non-Empty"::: ) ($#v11_pencil_1 :::"TopStruct-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ) & (Bool (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ) & (Bool "not" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "A")) ($#k1_pencil_1 :::"."::: ) (Set (Var "i"))) "is" ($#v3_pencil_1 :::"void"::: ) ))) ")" ) ")" )) "holds" (Bool (Set ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A"))) "is" ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ))) ; registrationlet "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "A" be ($#v15_pencil_1 :::"PLS-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Const "I")); cluster (Set ($#k5_pencil_1 :::"Segre_Product"::: ) "A") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#~v4_pencil_1 "non" ($#v4_pencil_1 :::"degenerated"::: ) ) ($#v5_pencil_1 :::"with_non_trivial_blocks"::: ) ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ; end; theorem :: PENCIL_1:19 (Bool "for" (Set (Var "T")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "T")) "st" (Bool (Bool (Set (Var "S")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) "holds" (Bool "(" (Bool (Set (Var "S")) "is" ($#v2_pencil_1 :::"strong"::: ) ) & (Bool (Set (Var "S")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) ")" ))) ; theorem :: PENCIL_1:20 (Bool "for" (Set (Var "S")) "being" ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "L")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "L")) ($#r1_hidden :::"="::: ) (Set (Var "l")))) "holds" (Bool (Set (Var "L")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) )))) ; theorem :: PENCIL_1:21 (Bool "for" (Set (Var "S")) "being" ($#l1_pre_topc :::"TopStruct"::: ) (Bool "for" (Set (Var "l")) "being" ($#m1_subset_1 :::"Block":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "L")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "L")) ($#r1_hidden :::"="::: ) (Set (Var "l")))) "holds" (Bool (Set (Var "L")) "is" ($#v2_pencil_1 :::"strong"::: ) )))) ; theorem :: PENCIL_1:22 (Bool "for" (Set (Var "S")) "being" ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#l1_pre_topc :::"TopStruct"::: ) "holds" (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set (Var "S"))) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) )) ; theorem :: PENCIL_1:23 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "A")))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "A"))))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_pencil_1 :::"indx"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "x")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "y")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))))))) ; theorem :: PENCIL_1:24 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v15_pencil_1 :::"PLS-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Block":::) "of" (Set "(" ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A")) ")" )) "iff" (Bool "ex" (Set (Var "L")) "being" ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m3_pboole :::"ManySortedSubset"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "A"))) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "L")))) & (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_pencil_1 :::"indx"::: ) (Set (Var "L")) ")" )) "is" ($#m1_subset_1 :::"Block":::) "of" (Set "(" (Set (Var "A")) ($#k2_pencil_1 :::"."::: ) (Set "(" ($#k3_pencil_1 :::"indx"::: ) (Set (Var "L")) ")" ) ")" )) ")" )) ")" )))) ; theorem :: PENCIL_1:25 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v15_pencil_1 :::"PLS-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "P")) "being" ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool (Set (Var "P")) "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A")) ")" ))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" (Set (Var "A")) ($#k2_pencil_1 :::"."::: ) (Set (Var "i")) ")" ) "holds" (Bool (Set (Set (Var "P")) ($#k2_funct_7 :::"+*"::: ) "(" (Set (Var "i")) "," (Set (Var "p")) ")" ) "is" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A")) ")" ))))))) ; theorem :: PENCIL_1:26 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) "st" (Bool (Bool (Num 2) ($#r1_ordinal1 :::"c="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set "(" (Set "(" ($#k4_card_3 :::"product"::: ) (Set (Var "A")) ")" ) ($#k3_xboole_0 :::"/\"::: ) (Set "(" ($#k4_card_3 :::"product"::: ) (Set (Var "B")) ")" ) ")" )))) "holds" (Bool "(" (Bool (Set ($#k3_pencil_1 :::"indx"::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set ($#k3_pencil_1 :::"indx"::: ) (Set (Var "B")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r1_hidden :::"<>"::: ) (Set ($#k3_pencil_1 :::"indx"::: ) (Set (Var "A"))))) "holds" (Bool (Set (Set (Var "A")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))) ")" ) ")" ))) ; theorem :: PENCIL_1:27 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "N")) "being" ($#~v1_zfmisc_1 "non" ($#v1_zfmisc_1 :::"trivial"::: ) ) ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "A")) ($#k2_funct_7 :::"+*"::: ) "(" (Set "(" ($#k3_pencil_1 :::"indx"::: ) (Set (Var "A")) ")" ) "," (Set (Var "N")) ")" ) "is" ($#v16_pencil_1 :::"Segre-like"::: ) ) & (Bool (Bool "not" (Set (Set (Var "A")) ($#k2_funct_7 :::"+*"::: ) "(" (Set "(" ($#k3_pencil_1 :::"indx"::: ) (Set (Var "A")) ")" ) "," (Set (Var "N")) ")" ) "is" ($#v13_pencil_1 :::"trivial-yielding"::: ) )) ")" )))) ; theorem :: PENCIL_1:28 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v3_pencil_1 "non" ($#v3_pencil_1 :::"void"::: ) ) ($#v6_pencil_1 :::"identifying_close_blocks"::: ) ($#v8_pencil_1 :::"without_isolated_points"::: ) ($#l1_pre_topc :::"TopStruct"::: ) "st" (Bool (Bool (Set (Var "S")) "is" ($#v10_pencil_1 :::"strongly_connected"::: ) )) "holds" (Bool (Set (Var "S")) "is" ($#v9_pencil_1 :::"connected"::: ) )) ; theorem :: PENCIL_1:29 (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#v15_pencil_1 :::"PLS-yielding"::: ) ($#m1_hidden :::"ManySortedSet":::) "of" (Set (Var "I")) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k5_pencil_1 :::"Segre_Product"::: ) (Set (Var "A")) ")" ) "holds" (Bool "(" (Bool "(" (Bool (Bool "not" (Set (Var "S")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) )) & (Bool (Set (Var "S")) "is" ($#v2_pencil_1 :::"strong"::: ) ) & (Bool (Set (Var "S")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) ")" ) "iff" (Bool "ex" (Set (Var "B")) "being" ($#~v13_pencil_1 "non" ($#v13_pencil_1 :::"trivial-yielding"::: ) ) ($#v16_pencil_1 :::"Segre-like"::: ) ($#m3_pboole :::"ManySortedSubset"::: ) "of" (Set ($#k12_pralg_1 :::"Carrier"::: ) (Set (Var "A"))) "st" (Bool "(" (Bool (Set (Var "S")) ($#r1_hidden :::"="::: ) (Set ($#k4_card_3 :::"product"::: ) (Set (Var "B")))) & (Bool "(" "for" (Set (Var "C")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "A")) ($#k2_pencil_1 :::"."::: ) (Set "(" ($#k3_pencil_1 :::"indx"::: ) (Set (Var "B")) ")" ) ")" ) "st" (Bool (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_pencil_1 :::"indx"::: ) (Set (Var "B")) ")" )))) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v2_pencil_1 :::"strong"::: ) ) & (Bool (Set (Var "C")) "is" ($#v1_pencil_1 :::"closed_under_lines"::: ) ) ")" ) ")" ) ")" )) ")" )))) ;