:: BSPACE semantic presentation begin definitionlet "S" be ($#l1_struct_0 :::"1-sorted"::: ) ; func :::"<*>"::: "S" -> ($#m2_finseq_1 :::"FinSequence":::) "of" "S" equals :: BSPACE:def 1 (Set ($#k6_finseq_1 :::"<*>"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) "S" ")" )); end; :: deftheorem defines :::"<*>"::: BSPACE:def 1 : (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) "holds" (Bool (Set ($#k1_bspace :::"<*>"::: ) (Set (Var "S"))) ($#r1_hidden :::"="::: ) (Set ($#k6_finseq_1 :::"<*>"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "S")) ")" )))); theorem :: BSPACE:1 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "S")) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "S")))))) ; theorem :: BSPACE:2 (Bool "for" (Set (Var "S")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_hidden :::"FinSequence":::) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "S"))) ")" )) "holds" (Bool (Set (Var "p")) "is" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set (Var "S"))))) ; scheme :: BSPACE:sch 1 IndSeqS{ F1() -> ($#l1_struct_0 :::"1-sorted"::: ) , P1[ ($#m1_hidden :::"set"::: ) ] } : (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set F1 "(" ")" ) "holds" (Bool P1[(Set (Var "p"))])) provided (Bool P1[(Set ($#k1_bspace :::"<*>"::: ) (Set F1 "(" ")" ))]) and (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set F1 "(" ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) "st" (Bool (Bool P1[(Set (Var "p"))])) "holds" (Bool P1[(Set (Set (Var "p")) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set (Var "x")) ($#k9_finseq_1 :::"*>"::: ) ))]))) proof end; begin definitionfunc :::"Z_2"::: -> ($#l6_algstr_0 :::"Field":::) equals :: BSPACE:def 2 (Set ($#k9_int_3 :::"INT.Ring"::: ) (Num 2)); end; :: deftheorem defines :::"Z_2"::: BSPACE:def 2 : (Bool (Set ($#k2_bspace :::"Z_2"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k9_int_3 :::"INT.Ring"::: ) (Num 2))); theorem :: BSPACE:3 (Bool (Set ($#k2_struct_0 :::"[#]"::: ) (Set ($#k2_bspace :::"Z_2"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k2_tarski :::"{"::: ) (Set ($#k1_xboole_0 :::"0"::: ) ) "," (Num 1) ($#k2_tarski :::"}"::: ) )) ; theorem :: BSPACE:4 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) "or" (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Num 1)) ")" )) ; theorem :: BSPACE:5 (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ; theorem :: BSPACE:6 (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )) ($#r1_hidden :::"="::: ) (Num 1)) ; theorem :: BSPACE:7 (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ; theorem :: BSPACE:8 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) "iff" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ")" )) ; begin definitionlet "X", "x" be ($#m1_hidden :::"set"::: ) ; func "X" :::"@"::: "x" -> ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) equals :: BSPACE:def 3 (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )) if (Bool "x" ($#r2_hidden :::"in"::: ) "X") otherwise (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )); end; :: deftheorem defines :::"@"::: BSPACE:def 3 : (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "implies" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))) "implies" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ")" ")" )); theorem :: BSPACE:9 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) "iff" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" )) ; theorem :: BSPACE:10 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) "iff" (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) ")" )) ; theorem :: BSPACE:11 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) "iff" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ")" )) ; theorem :: BSPACE:12 (Bool "for" (Set (Var "X")) "," (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "y")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "iff" (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ) ")" )) ; theorem :: BSPACE:13 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "," (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "Y")) ($#k3_bspace :::"@"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) "iff" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) ")" ) ")" )) ; theorem :: BSPACE:14 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k1_xboole_0 :::"{}"::: ) ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))) ; theorem :: BSPACE:15 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "u")) ($#k5_subset_1 :::"\+\"::: ) (Set (Var "v")) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "u")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "v")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" )))))) ; theorem :: BSPACE:16 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "X")) ($#r1_hidden :::"="::: ) (Set (Var "Y"))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set (Var "X")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "Y")) ($#k3_bspace :::"@"::: ) (Set (Var "x"))))) ")" )) ; begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ); let "c" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "X")); func "a" :::"\*\"::: "c" -> ($#m1_subset_1 :::"Subset":::) "of" "X" equals :: BSPACE:def 4 "c" if (Bool "a" ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) (Set ($#k1_subset_1 :::"{}"::: ) "X") if (Bool "a" ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))) ; end; :: deftheorem defines :::"\*\"::: BSPACE:def 4 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))) "implies" (Bool (Set (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Var "c"))) ")" & "(" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))) "implies" (Bool (Set (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_subset_1 :::"{}"::: ) (Set (Var "X")))) ")" ")" )))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"bspace-sum"::: "X" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "X" ")" ) means :: BSPACE:def 5 (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "holds" (Bool (Set it ($#k5_binop_1 :::"."::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_subset_1 :::"\+\"::: ) (Set (Var "d"))))); end; :: deftheorem defines :::"bspace-sum"::: BSPACE:def 5 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_bspace :::"bspace-sum"::: ) (Set (Var "X")))) "iff" (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b2")) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "c")) "," (Set (Var "d")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "c")) ($#k5_subset_1 :::"\+\"::: ) (Set (Var "d"))))) ")" ))); theorem :: BSPACE:17 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set "(" (Set (Var "c")) ($#k5_subset_1 :::"\+\"::: ) (Set (Var "d")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c")) ")" ) ($#k5_subset_1 :::"\+\"::: ) (Set "(" (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set (Var "d")) ")" )))))) ; theorem :: BSPACE:18 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k4_bspace :::"\*\"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c")) ")" ) ($#k5_subset_1 :::"\+\"::: ) (Set "(" (Set (Var "b")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c")) ")" )))))) ; theorem :: BSPACE:19 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ) ($#k4_bspace :::"\*\"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Var "c"))))) ; theorem :: BSPACE:20 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set "(" (Set (Var "b")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "b")) ")" ) ($#k4_bspace :::"\*\"::: ) (Set (Var "c"))))))) ; definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"bspace-scalar-mult"::: "X" -> ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_bspace :::"Z_2"::: ) )) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "X" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "X" ")" ) means :: BSPACE:def 6 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "holds" (Bool (Set it ($#k2_binop_1 :::"."::: ) "(" (Set (Var "a")) "," (Set (Var "c")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c")))))); end; :: deftheorem defines :::"bspace-scalar-mult"::: BSPACE:def 6 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set ($#k2_bspace :::"Z_2"::: ) )) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k6_bspace :::"bspace-scalar-mult"::: ) (Set (Var "X")))) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "b2")) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "a")) "," (Set (Var "c")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_bspace :::"\*\"::: ) (Set (Var "c")))))) ")" ))); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"bspace"::: "X" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectsp_1 :::"VectSpStr"::: ) "over" (Set ($#k2_bspace :::"Z_2"::: ) ) equals :: BSPACE:def 7 (Set ($#g1_vectsp_1 :::"VectSpStr"::: ) "(#" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) "X" ")" ) "," (Set "(" ($#k5_bspace :::"bspace-sum"::: ) "X" ")" ) "," (Set "(" ($#k1_subset_1 :::"{}"::: ) "X" ")" ) "," (Set "(" ($#k6_bspace :::"bspace-scalar-mult"::: ) "X" ")" ) "#)" ); end; :: deftheorem defines :::"bspace"::: BSPACE:def 7 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#g1_vectsp_1 :::"VectSpStr"::: ) "(#" (Set "(" ($#k1_zfmisc_1 :::"bool"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k5_bspace :::"bspace-sum"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k1_subset_1 :::"{}"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k6_bspace :::"bspace-scalar-mult"::: ) (Set (Var "X")) ")" ) "#)" ))); theorem :: BSPACE:21 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v2_rlvect_1 :::"Abelian"::: ) )) ; theorem :: BSPACE:22 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v3_rlvect_1 :::"add-associative"::: ) )) ; theorem :: BSPACE:23 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v4_rlvect_1 :::"right_zeroed"::: ) )) ; theorem :: BSPACE:24 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v13_algstr_0 :::"right_complementable"::: ) )) ; theorem :: BSPACE:25 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "y")) ")" )))))) ; theorem :: BSPACE:26 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "b")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "x")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "x")) ")" )))))) ; theorem :: BSPACE:27 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "b")) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k4_vectsp_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "x")) ")" )))))) ; theorem :: BSPACE:28 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set (Set "(" ($#k1_group_1 :::"1_"::: ) (Set ($#k2_bspace :::"Z_2"::: ) ) ")" ) ($#k4_vectsp_1 :::"*"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Var "x"))))) ; theorem :: BSPACE:29 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v8_vectsp_1 :::"vector-distributive"::: ) ) & (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v9_vectsp_1 :::"scalar-distributive"::: ) ) & (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v10_vectsp_1 :::"scalar-associative"::: ) ) & (Bool (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) "is" ($#v11_vectsp_1 :::"scalar-unital"::: ) ) ")" )) ; registrationlet "X" be ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_bspace :::"bspace"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v8_vectsp_1 :::"vector-distributive"::: ) ($#v9_vectsp_1 :::"scalar-distributive"::: ) ($#v10_vectsp_1 :::"scalar-associative"::: ) ($#v11_vectsp_1 :::"scalar-unital"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ; end; begin definitionlet "X" be ($#m1_hidden :::"set"::: ) ; func :::"singletons"::: "X" -> ($#m1_hidden :::"set"::: ) equals :: BSPACE:def 8 "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Subset":::) "of" "X" : (Bool (Set (Var "f")) "is" (Num 1) ($#v3_card_1 :::"-element"::: ) ) "}" ; end; :: deftheorem defines :::"singletons"::: BSPACE:def 8 : (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k8_bspace :::"singletons"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) : (Bool (Set (Var "f")) "is" (Num 1) ($#v3_card_1 :::"-element"::: ) ) "}" )); definitionlet "X" be ($#m1_hidden :::"set"::: ) ; :: original: :::"singletons"::: redefine func :::"singletons"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) "X" ")" ); end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k8_bspace :::"singletons"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: BSPACE:30 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) "is" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k9_bspace :::"singletons"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "f")) "is" (Num 1) ($#v3_card_1 :::"-element"::: ) ))) ; definitionlet "F" be ($#l6_algstr_0 :::"Field":::); let "V" be ($#l1_vectsp_1 :::"VectSp":::) "of" (Set (Const "F")); let "l" be ($#m1_vectsp_6 :::"Linear_Combination"::: ) "of" (Set (Const "V")); let "x" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "V")); :: original: :::"."::: redefine func "l" :::"."::: "x" -> ($#m1_subset_1 :::"Element":::) "of" "F"; end; definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "s" be ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Const "X")) ")" ); let "x" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); func "s" :::"@"::: "x" -> ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) means :: BSPACE:def 9 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "s")) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "s"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "s" ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k3_bspace :::"@"::: ) "x")) ")" ) ")" ); end; :: deftheorem defines :::"@"::: BSPACE:def 9 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) (Bool "for" (Set (Var "b4")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set ($#k2_bspace :::"Z_2"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "s")) ($#k11_bspace :::"@"::: ) (Set (Var "x")))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s")))) & (Bool "(" "for" (Set (Var "j")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "j")) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x")))) ")" ) ")" ) ")" ))))); theorem :: BSPACE:31 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" ($#k1_bspace :::"<*>"::: ) (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) ")" ) ($#k11_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_bspace :::"<*>"::: ) (Set ($#k2_bspace :::"Z_2"::: ) ))))) ; theorem :: BSPACE:32 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v")) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "u")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "v")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" )))))) ; theorem :: BSPACE:33 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" (Set (Var "s")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "f")) ($#k12_finseq_1 :::"*>"::: ) ) ")" ) ($#k11_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "s")) ($#k11_bspace :::"@"::: ) (Set (Var "x")) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "f")) ($#k3_bspace :::"@"::: ) (Set (Var "x")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))))))) ; theorem :: BSPACE:34 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "s")) "being" ($#m2_finseq_1 :::"FinSequence":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "holds" (Bool (Set (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "s")) ")" ) ($#k3_bspace :::"@"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" (Set (Var "s")) ($#k11_bspace :::"@"::: ) (Set (Var "x")) ")" )))))) ; theorem :: BSPACE:35 (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "l")) "being" ($#m1_vectsp_6 :::"Linear_Combination"::: ) "of" (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_vectsp_6 :::"Carrier"::: ) (Set (Var "l"))))) "holds" (Bool (Set (Set (Var "l")) ($#k10_bspace :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_group_1 :::"1_"::: ) (Set ($#k2_bspace :::"Z_2"::: ) )))))) ; registrationlet "X" be ($#v1_xboole_0 :::"empty"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k8_bspace :::"singletons"::: ) "X") -> ($#v1_xboole_0 :::"empty"::: ) ; end; theorem :: BSPACE:36 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_bspace :::"singletons"::: ) (Set (Var "X"))) "is" ($#v1_vectsp_7 :::"linearly-independent"::: ) )) ; theorem :: BSPACE:37 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) "st" (Bool (Bool "ex" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set (Var "x")) ($#k1_tarski :::"}"::: ) )) ")" ))) "holds" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k9_bspace :::"singletons"::: ) (Set (Var "X")))))) ; theorem :: BSPACE:38 (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "ex" (Set (Var "l")) "being" ($#m2_vectsp_6 :::"Linear_Combination"::: ) "of" (Set ($#k9_bspace :::"singletons"::: ) (Set (Var "X"))) "st" (Bool (Set ($#k4_vectsp_6 :::"Sum"::: ) (Set (Var "l"))) ($#r1_hidden :::"="::: ) (Set (Var "A")))))) ; theorem :: BSPACE:39 (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_vectsp_7 :::"Lin"::: ) (Set "(" ($#k9_bspace :::"singletons"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))))) ; theorem :: BSPACE:40 (Bool "for" (Set (Var "X")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k9_bspace :::"singletons"::: ) (Set (Var "X"))) "is" ($#m1_vectsp_7 :::"Basis"::: ) "of" (Set ($#k7_bspace :::"bspace"::: ) (Set (Var "X"))))) ; registrationlet "X" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k8_bspace :::"singletons"::: ) "X") -> ($#v1_finset_1 :::"finite"::: ) ; end; registrationlet "X" be ($#v1_finset_1 :::"finite"::: ) ($#m1_hidden :::"set"::: ) ; cluster (Set ($#k7_bspace :::"bspace"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_matrlin :::"finite-dimensional"::: ) ; end; theorem :: BSPACE:41 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k9_bspace :::"singletons"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_card_1 :::"card"::: ) (Set (Var "X"))))) ; theorem :: BSPACE:42 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set ($#k1_card_1 :::"card"::: ) (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set "(" ($#k7_bspace :::"bspace"::: ) (Set (Var "X")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_card_2 :::"exp"::: ) "(" (Num 2) "," (Set "(" ($#k1_card_1 :::"card"::: ) (Set (Var "X")) ")" ) ")" ))) ; theorem :: BSPACE:43 (Bool (Set ($#k1_vectsp_9 :::"dim"::: ) (Set "(" ($#k7_bspace :::"bspace"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"0"::: ) )) ;