:: RMOD_2 semantic presentation begin definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "V1" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "V")); attr "V1" is :::"linearly-closed"::: means :: RMOD_2:def 1 (Bool "(" (Bool "(" "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#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" ($#m1_subset_1 :::"Scalar":::) "of" "R" (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" "V" "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) "V1")) "holds" (Bool (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) "V1")) ")" ) ")" ); end; :: deftheorem defines :::"linearly-closed"::: RMOD_2:def 1 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "V1")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) ) "iff" (Bool "(" (Bool "(" "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#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" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) "holds" (Bool (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))) ")" ) ")" ) ")" )))); theorem :: RMOD_2:1 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "being" ($#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" ($#v1_rmod_2 :::"linearly-closed"::: ) )) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1")))))) ; theorem :: RMOD_2:2 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) )) "holds" (Bool "for" (Set (Var "v")) "being" ($#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"))) ($#r2_hidden :::"in"::: ) (Set (Var "V1"))))))) ; theorem :: RMOD_2:3 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) )) "holds" (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#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 :: RMOD_2:4 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) ))) ; theorem :: RMOD_2:5 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (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_rmod_2 :::"linearly-closed"::: ) )))) ; theorem :: RMOD_2:6 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (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_rmod_2 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V2")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V3")) ($#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" ($#v1_rmod_2 :::"linearly-closed"::: ) )))) ; theorem :: RMOD_2:7 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "," (Set (Var "V2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) ) & (Bool (Set (Var "V2")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) )) "holds" (Bool (Set (Set (Var "V1")) ($#k9_subset_1 :::"/\"::: ) (Set (Var "V2"))) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) )))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); mode :::"Submodule"::: "of" "V" -> ($#l1_vectsp_2 :::"RightMod":::) "of" "R" means :: RMOD_2:def 2 (Bool "(" (Bool (Set "the" ($#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") ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set "the" ($#u1_vectsp_2 :::"rmult"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_vectsp_2 :::"rmult"::: ) "of" "V") ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") ($#k2_zfmisc_1 :::":]"::: ) ))) ")" ); end; :: deftheorem defines :::"Submodule"::: RMOD_2:def 2 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "," (Set (Var "b3")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V"))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V")))) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set "the" ($#u1_vectsp_2 :::"rmult"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_vectsp_2 :::"rmult"::: ) "of" (Set (Var "V"))) ($#k5_relat_1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ))) ")" ) ")" ))); theorem :: RMOD_2:8 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W1"))) & (Bool (Set (Var "W1")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "W2")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2"))))))) ; theorem :: RMOD_2:9 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set (Var "V"))))))) ; theorem :: RMOD_2:10 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "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 :: RMOD_2:11 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))))))) ; theorem :: RMOD_2:12 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W2"))))))) ; theorem :: RMOD_2:13 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "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 :: RMOD_2:14 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Var "v")))) "holds" (Bool (Set (Set (Var "w")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")))))))))) ; theorem :: RMOD_2:15 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "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 ($#k4_algstr_0 :::"-"::: ) (Set (Var "w"))))))))) ; theorem :: RMOD_2:16 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "w1")) "," (Set (Var "w2")) "being" ($#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")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u"))))))))) ; theorem :: RMOD_2:17 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))) ; theorem :: RMOD_2:18 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W1"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W2")))))) ; theorem :: RMOD_2:19 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "W"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "V")))))) ; theorem :: RMOD_2:20 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "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 :: RMOD_2:21 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))))))) ; theorem :: RMOD_2:22 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "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"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: RMOD_2:23 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#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"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))))))) ; theorem :: RMOD_2:24 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set (Var "V")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V"))))) ; theorem :: RMOD_2:25 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "X")) "," (Set (Var "V")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "V")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")))) "holds" (Bool (Set (Var "V")) ($#r1_hidden :::"="::: ) (Set (Var "X"))))) ; registrationlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_2 :::"strict"::: ) ($#v4_vectsp_2 :::"RightMod-like"::: ) for ($#m1_rmod_2 :::"Submodule"::: ) "of" "V"; end; theorem :: RMOD_2:26 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "," (Set (Var "X")) "," (Set (Var "Y")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "V")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "X"))) & (Bool (Set (Var "X")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "Y")))) "holds" (Bool (Set (Var "V")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "Y"))))) ; theorem :: RMOD_2:27 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#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" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "W2")))))) ; theorem :: RMOD_2:28 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "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_rmod_2 :::"Submodule"::: ) "of" (Set (Var "W2")))))) ; theorem :: RMOD_2:29 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "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 :: RMOD_2:30 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "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 :: RMOD_2:31 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#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")) ($#r1_hidden :::"="::: ) (Set (Var "V")))))) ; theorem :: RMOD_2:32 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool "(" "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" )) "holds" (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set (Var "V")))))) ; theorem :: RMOD_2:33 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Var "V1")))) "holds" (Bool (Set (Var "V1")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) ))))) ; theorem :: RMOD_2:34 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (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_rmod_2 :::"linearly-closed"::: ) )) "holds" (Bool "ex" (Set (Var "W")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); func :::"(0)."::: "V" -> ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" "V" means :: RMOD_2:def 3 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "V" ")" ) ($#k6_domain_1 :::"}"::: ) )); end; :: deftheorem defines :::"(0)."::: RMOD_2:def 3 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b3")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V")))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ")" )))); definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); func :::"(Omega)."::: "V" -> ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" "V" equals :: RMOD_2:def 4 (Set ($#g1_vectsp_2 :::"RightModStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "V") "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" "V") "," (Set "the" ($#u1_vectsp_2 :::"rmult"::: ) "of" "V") "#)" ); end; :: deftheorem defines :::"(Omega)."::: RMOD_2:def 4 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k2_rmod_2 :::"(Omega)."::: ) (Set (Var "V"))) ($#r1_hidden :::"="::: ) (Set ($#g1_vectsp_2 :::"RightModStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "V"))) "," (Set "the" ($#u1_vectsp_2 :::"rmult"::: ) "of" (Set (Var "V"))) "#)" )))); theorem :: RMOD_2:35 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r1_struct_0 :::"in"::: ) (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V")))) "iff" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V")))) ")" )))) ; theorem :: RMOD_2:36 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V"))))))) ; theorem :: RMOD_2:37 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "W2"))))))) ; theorem :: RMOD_2:38 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "W"))) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")))))) ; theorem :: RMOD_2:39 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V"))) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "W")))))) ; theorem :: RMOD_2:40 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "W1"))) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "W2")))))) ; theorem :: RMOD_2:41 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set (Var "V")) "is" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set ($#k2_rmod_2 :::"(Omega)."::: ) (Set (Var "V")))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "v" be ($#m1_subset_1 :::"Vector":::) "of" (Set (Const "V")); let "W" be ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Const "V")); func "v" :::"+"::: "W" -> ($#m1_subset_1 :::"Subset":::) "of" "V" equals :: RMOD_2:def 5 "{" (Set "(" "v" ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) where u "is" ($#m1_subset_1 :::"Vector":::) "of" "V" : (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) "W") "}" ; end; :: deftheorem defines :::"+"::: RMOD_2:def 5 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (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 "R" be ($#l6_algstr_0 :::"Ring":::); let "V" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "W" be ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Const "V")); mode :::"Coset"::: "of" "W" -> ($#m1_subset_1 :::"Subset":::) "of" "V" means :: RMOD_2:def 6 (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" "V" "st" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) "W"))); end; :: deftheorem defines :::"Coset"::: RMOD_2:def 6 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b4")) "is" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W"))) "iff" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))))) ")" ))))); theorem :: RMOD_2:42 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) "iff" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")))) ")" )) ")" )))))) ; theorem :: RMOD_2:43 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" ))))) ; theorem :: RMOD_2:44 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))))))) ; theorem :: RMOD_2:45 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))))) ; theorem :: RMOD_2:46 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set "(" ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ))))) ; theorem :: RMOD_2:47 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set "(" ($#k2_rmod_2 :::"(Omega)."::: ) (Set (Var "V")) ")" )) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))))))) ; theorem :: RMOD_2:48 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "V"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" ))))) ; theorem :: RMOD_2:49 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" ))))) ; theorem :: RMOD_2:50 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set "(" (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a")) ")" ) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))))))))) ; theorem :: RMOD_2:51 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "u")) ")" ) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) ")" ))))) ; theorem :: RMOD_2:52 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) "iff" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u")) ")" ) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) ")" ))))) ; theorem :: RMOD_2:53 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "u")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Set (Var "u")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) ")" ))))) ; theorem :: RMOD_2:54 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v1")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v2")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool (Set (Set (Var "v1")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v2")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))))))) ; theorem :: RMOD_2:55 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Scalar":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set (Set (Var "v")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))))))))) ; theorem :: RMOD_2:56 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "v")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))))))) ; theorem :: RMOD_2:57 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "u")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "v"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" ))))) ; theorem :: RMOD_2:58 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) "iff" (Bool (Set (Var "u")) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" ))))) ; theorem :: RMOD_2:59 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) "iff" (Bool "ex" (Set (Var "v1")) "being" ($#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")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v1")))) ")" )) ")" ))))) ; theorem :: RMOD_2:60 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) & (Bool (Set (Var "v2")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) ")" )) "iff" (Bool (Set (Set (Var "v1")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v2"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" ))))) ; theorem :: RMOD_2:61 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))))) "holds" (Bool "ex" (Set (Var "v1")) "being" ($#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"))) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" )))))) ; theorem :: RMOD_2:62 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rmod_2 :::"+"::: ) (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 :: RMOD_2:63 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W2")))) "iff" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))) ")" ))))) ; theorem :: RMOD_2:64 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W1"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W2"))))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2"))))))) ; theorem :: RMOD_2:65 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "ex" (Set (Var "C")) "being" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "C")))))))) ; theorem :: RMOD_2:66 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set (Var "C")) "is" ($#v1_rmod_2 :::"linearly-closed"::: ) ) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W")))) ")" ))))) ; theorem :: RMOD_2:67 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W1")) "," (Set (Var "W2")) "being" ($#v2_vectsp_2 :::"strict"::: ) ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C1")) "being" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W1")) (Bool "for" (Set (Var "C2")) "being" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W2")) "st" (Bool (Bool (Set (Var "C1")) ($#r1_hidden :::"="::: ) (Set (Var "C2")))) "holds" (Bool (Set (Var "W1")) ($#r1_hidden :::"="::: ) (Set (Var "W2")))))))) ; theorem :: RMOD_2:68 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) ) "is" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V"))))))) ; theorem :: RMOD_2:69 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set ($#k1_rmod_2 :::"(0)."::: ) (Set (Var "V"))))) "holds" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) "st" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "v")) ($#k6_domain_1 :::"}"::: ) )))))) ; theorem :: RMOD_2:70 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) "is" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W")))))) ; theorem :: RMOD_2:71 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "is" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set ($#k2_rmod_2 :::"(Omega)."::: ) (Set (Var "V")))))) ; theorem :: RMOD_2:72 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "V1")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "V")) "st" (Bool (Bool (Set (Var "V1")) "is" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set ($#k2_rmod_2 :::"(Omega)."::: ) (Set (Var "V"))))) "holds" (Bool (Set (Var "V1")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))))))) ; theorem :: RMOD_2:73 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rmod_2 :::"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 :: RMOD_2:74 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W")) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "C"))) "iff" (Bool (Set (Var "C")) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k3_rmod_2 :::"+"::: ) (Set (Var "W")))) ")" )))))) ; theorem :: RMOD_2:75 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool (Bool (Set (Var "u")) ($#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"))) ($#r1_hidden :::"="::: ) (Set (Var "v"))) ")" ))))))) ; theorem :: RMOD_2:76 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "C")) "being" ($#m2_rmod_2 :::"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 :: RMOD_2:77 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "v1")) "," (Set (Var "v2")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) "holds" (Bool "(" (Bool "ex" (Set (Var "C")) "being" ($#m2_rmod_2 :::"Coset"::: ) "of" (Set (Var "W")) "st" (Bool "(" (Bool (Set (Var "v1")) ($#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"))) ($#r1_struct_0 :::"in"::: ) (Set (Var "W"))) ")" ))))) ; theorem :: RMOD_2:78 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "V")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Vector":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "W")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Var "V")) (Bool "for" (Set (Var "B")) "," (Set (Var "C")) "being" ($#m2_rmod_2 :::"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")))))))) ;