:: RLSUB_1 semantic presentation begin definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "V1" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "V1" is :::"linearly-closed"::: means :: RLSUB_1:def 1 (Bool "(" (Bool "(" "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "V1") & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "V1")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) "V1") ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "V1")) "holds" (Bool (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) "V1")) ")" ) ")" ); end; :: deftheorem defines :::"linearly-closed"::: RLSUB_1:def 1 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ) "iff" (Bool "(" (Bool "(" "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) ")" ) ")" ) ")" ))); theorem :: RLSUB_1:1 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))))) ; theorem :: RLSUB_1:2 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )) "holds" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))))) ; theorem :: RLSUB_1:3 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )) "holds" (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))))) ; theorem :: RLSUB_1:4 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )) ; theorem :: RLSUB_1:5 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set (Var "V1")))) "holds" (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ))) ; theorem :: RLSUB_1:6 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "," (Set (Var "V2")) "," (Set (Var "V3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V2")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V3")) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) where v, u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "V2"))) ")" ) "}" )) "holds" (Bool (Set (Var "V3")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ))) ; theorem :: RLSUB_1:7 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "," (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V2")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )) "holds" (Bool (Set (Set (Var "V1")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "V2"))) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ))) ; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); mode :::"Subspace"::: "of" "V" -> ($#l1_rlvect_1 :::"RealLinearSpace":::) means :: RLSUB_1:def 2 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V")) & (Bool (Set ($#k4_struct_0 :::"0."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "V")) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "V") ($#k1_realset1 :::"||"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it))) & (Bool (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" "V") ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#k2_zfmisc_1 :::":]"::: ) ))) ")" ); end; :: deftheorem defines :::"Subspace"::: RLSUB_1:def 2 : (Bool "for" (Set (Var "V")) "," (Set (Var "b2")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V"))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) ($#k1_realset1 :::"||"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))))) & (Bool (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#k2_zfmisc_1 :::":]"::: ) ))) ")" ) ")" )); theorem :: RLSUB_1:8 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W1"))) & (Bool (Set (Var "W1")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "W2")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2")))))) ; theorem :: RLSUB_1:9 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "V")))))) ; theorem :: RLSUB_1:10 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "holds" (Bool (Set (Var "w")) "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")))))) ; theorem :: RLSUB_1:11 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))))) ; theorem :: RLSUB_1:12 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W2")))))) ; theorem :: RLSUB_1:13 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w1")) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set (Var "w2")) ($#r1_hidden :::"="::: ) (Set (Var "u")))) "holds" (Bool (Set (Set (Var "w1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "w2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")))))))) ; theorem :: RLSUB_1:14 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v"))))))))) ; theorem :: RLSUB_1:15 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "w")))))))) ; theorem :: RLSUB_1:16 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w1")) ($#r1_hidden :::"="::: ) (Set (Var "v"))) & (Bool (Set (Var "w2")) ($#r1_hidden :::"="::: ) (Set (Var "u")))) "holds" (Bool (Set (Set (Var "w1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "w2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u")))))))) ; theorem :: RLSUB_1:17 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))) ; theorem :: RLSUB_1:18 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W1"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2"))))) ; theorem :: RLSUB_1:19 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "V"))))) ; theorem :: RLSUB_1:20 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: RLSUB_1:21 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: RLSUB_1:22 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: RLSUB_1:23 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: RLSUB_1:24 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "D")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "d1")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set (Var "D")) (Bool "for" (Set (Var "M")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set (Var "D")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "D")) "st" (Bool (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool (Set (Var "d1")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) ($#k1_realset1 :::"||"::: ) (Set (Var "V1")))) & (Bool (Set (Var "M")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set (Var "V1")) ($#k2_zfmisc_1 :::":]"::: ) )))) "holds" (Bool (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set (Var "D")) "," (Set (Var "d1")) "," (Set (Var "A")) "," (Set (Var "M")) "#)" ) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V"))))))))) ; theorem :: RLSUB_1:25 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set (Var "V")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")))) ; theorem :: RLSUB_1:26 (Bool "for" (Set (Var "V")) "," (Set (Var "X")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")))) "holds" (Bool (Set (Var "V")) ($#r1_hidden :::"="::: ) (Set (Var "X")))) ; theorem :: RLSUB_1:27 (Bool "for" (Set (Var "V")) "," (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "Y")))) "holds" (Bool (Set (Var "V")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "Y")))) ; theorem :: RLSUB_1:28 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W1"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W2"))))) "holds" (Bool (Set (Var "W1")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "W2"))))) ; theorem :: RLSUB_1:29 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W1")))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2"))) ")" )) "holds" (Bool (Set (Var "W1")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "W2"))))) ; registrationlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v12_algstr_0 :::"left_complementable"::: ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v14_algstr_0 :::"complementable"::: ) ($#v1_rlvect_1 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) bbbadV9_RLVECT_1() for ($#m1_rlsub_1 :::"Subspace"::: ) "of" "V"; end; theorem :: RLSUB_1:30 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W2"))))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))))) ; theorem :: RLSUB_1:31 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W1"))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2"))) ")" ) ")" )) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))))) ; theorem :: RLSUB_1:32 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))))) "holds" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V"))))) ; theorem :: RLSUB_1:33 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "V"))) ")" ) ")" )) "holds" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V"))))) ; theorem :: RLSUB_1:34 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Var "V1")))) "holds" (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )))) ; theorem :: RLSUB_1:35 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "V1")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )) "holds" (Bool "ex" (Set (Var "W")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))))) ; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); func :::"(0)."::: "V" -> ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" "V" means :: RLSUB_1:def 3 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "V" ")" ) ($#k6_domain_1 :::"}"::: ) )); end; :: deftheorem defines :::"(0)."::: RLSUB_1:def 3 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "b2")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V")))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ")" ))); definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); func :::"(Omega)."::: "V" -> ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" "V" equals :: RLSUB_1:def 4 (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" "V") "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "V") "," (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" "V") "#)" ); end; :: deftheorem defines :::"(Omega)."::: RLSUB_1:def 4 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "V"))) "#)" ))); theorem :: RLSUB_1:36 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V")))))) ; theorem :: RLSUB_1:37 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "W2")))))) ; theorem :: RLSUB_1:38 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "W"))) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V"))))) ; theorem :: RLSUB_1:39 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V"))) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "W"))))) ; theorem :: RLSUB_1:40 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "W1"))) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "W2"))))) ; theorem :: RLSUB_1:41 (Bool "for" (Set (Var "V")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set (Var "V")) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))))) ; definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "v" be ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Const "V")); let "W" be ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Const "V")); func "v" :::"+"::: "W" -> ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: RLSUB_1:def 5 "{" (Set "(" "v" ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" "V" : (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) "W") "}" ; end; :: deftheorem defines :::"+"::: RLSUB_1:def 5 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) where u "is" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) : (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "}" )))); definitionlet "V" be ($#l1_rlvect_1 :::"RealLinearSpace":::); let "W" be ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Const "V")); mode :::"Coset"::: "of" "W" -> ($#m1_subset_1 :::"Subset":::) "of" "V" means :: RLSUB_1:def 6 (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" "V" "st" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) "W"))); end; :: deftheorem defines :::"Coset"::: RLSUB_1:def 6 : (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W"))) "iff" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))) ")" )))); theorem :: RLSUB_1:42 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: RLSUB_1:43 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))))) ; theorem :: RLSUB_1:44 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))))) ; theorem :: RLSUB_1:45 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set "(" ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: RLSUB_1:46 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set "(" ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))))) ; theorem :: RLSUB_1:47 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:48 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:49 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))))))) ; theorem :: RLSUB_1:50 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: RLSUB_1:51 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:52 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:53 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:54 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "u")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Set (Var "u")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:55 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: RLSUB_1:56 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v1")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v2")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Set (Var "v1")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v2")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))))) ; theorem :: RLSUB_1:57 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: RLSUB_1:58 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Num 1)) & (Bool (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: RLSUB_1:59 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))))))) ; theorem :: RLSUB_1:60 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: RLSUB_1:61 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: RLSUB_1:62 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: RLSUB_1:63 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v1")))) ")" )) ")" )))) ; theorem :: RLSUB_1:64 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) "iff" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v1")))) ")" )) ")" )))) ; theorem :: RLSUB_1:65 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) ")" )) "iff" (Bool (Set (Set (Var "v1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v2"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: RLSUB_1:66 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))))) ; theorem :: RLSUB_1:67 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))))) ; theorem :: RLSUB_1:68 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W2")))) "iff" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))) ")" )))) ; theorem :: RLSUB_1:69 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W2"))))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2")))))) ; theorem :: RLSUB_1:70 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) ) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:71 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v1_rlvect_1 :::"strict"::: ) ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C1")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W1")) (Bool "for" (Set (Var "C2")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W2")) "st" (Bool (Bool (Set (Var "C1")) ($#r1_hidden :::"="::: ) (Set (Var "C2")))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))))))) ; theorem :: RLSUB_1:72 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V")))))) ; theorem :: RLSUB_1:73 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set ($#k1_rlsub_1 :::"(0)."::: ) (Set (Var "V"))))) "holds" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: RLSUB_1:74 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) "is" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W"))))) ; theorem :: RLSUB_1:75 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "is" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))))) ; theorem :: RLSUB_1:76 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set ($#k2_rlsub_1 :::"(Omega)."::: ) (Set (Var "V"))))) "holds" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))))) ; theorem :: RLSUB_1:77 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" )))) ; theorem :: RLSUB_1:78 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rlsub_1 :::"+"::: ) (Set (Var "W")))) ")" ))))) ; theorem :: RLSUB_1:79 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "v"))) ")" )))))) ; theorem :: RLSUB_1:80 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Set (Var "u")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v1"))) ($#r1_hidden :::"="::: ) (Set (Var "v"))) ")" )))))) ; theorem :: RLSUB_1:81 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool "ex" (Set (Var "C")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) ")" )) "iff" (Bool (Set (Set (Var "v1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v2"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )))) ; theorem :: RLSUB_1:82 (Bool "for" (Set (Var "V")) "being" ($#l1_rlvect_1 :::"RealLinearSpace":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "," (Set (Var "C")) "being" ($#m2_rlsub_1 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "B"))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))) "holds" (Bool (Set (Var "B")) ($#r1_hidden :::"="::: ) (Set (Var "C"))))))) ;