:: IDEAL_1 semantic presentation begin registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) for ($#l2_algstr_0 :::"addLoopStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; theorem :: IDEAL_1:1 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "v")) "," (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set ($#k2_finseq_4 :::"<*"::: ) (Set (Var "v")) "," (Set (Var "u")) ($#k2_finseq_4 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "u")))))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "F" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); attr "F" is :::"add-closed"::: means :: IDEAL_1:def 1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "F") & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) "F")) "holds" (Bool (Set (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) "F")); end; :: deftheorem defines :::"add-closed"::: IDEAL_1:def 1 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v1_ideal_1 :::"add-closed"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) ; let "F" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); attr "F" is :::"left-ideal"::: means :: IDEAL_1:def 2 (Bool "for" (Set (Var "p")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "F")) "holds" (Bool (Set (Set (Var "p")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) "F")); attr "F" is :::"right-ideal"::: means :: IDEAL_1:def 3 (Bool "for" (Set (Var "p")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "F")) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) "F")); end; :: deftheorem defines :::"left-ideal"::: IDEAL_1:def 2 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v2_ideal_1 :::"left-ideal"::: ) ) "iff" (Bool "for" (Set (Var "p")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Set (Var "p")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); :: deftheorem defines :::"right-ideal"::: IDEAL_1:def 3 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "F")) "is" ($#v3_ideal_1 :::"right-ideal"::: ) ) "iff" (Bool "for" (Set (Var "p")) "," (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "p"))) ($#r2_hidden :::"in"::: ) (Set (Var "F")))) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#l3_algstr_0 :::"multMagma"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R")); end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; mode Ideal of "L" is ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" "L"; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; mode RightIdeal of "L" is ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" "L"; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; mode LeftIdeal of "L" is ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" "L"; end; theorem :: IDEAL_1:2 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r2_hidden :::"in"::: ) (Set (Var "I"))))) ; theorem :: IDEAL_1:3 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))) ($#r2_hidden :::"in"::: ) (Set (Var "I"))))) ; theorem :: IDEAL_1:4 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v1_ideal_1 :::"add-closed"::: ) )) ; theorem :: IDEAL_1:5 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v2_ideal_1 :::"left-ideal"::: ) )) ; theorem :: IDEAL_1:6 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#v3_ideal_1 :::"right-ideal"::: ) )) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; cluster (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "L" ")" ) ($#k1_tarski :::"}"::: ) ) -> ($#v1_ideal_1 :::"add-closed"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "L"; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "L" ")" ) ($#k1_tarski :::"}"::: ) ) -> ($#v2_ideal_1 :::"left-ideal"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "L"; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "L" ")" ) ($#k1_tarski :::"}"::: ) ) -> ($#v3_ideal_1 :::"right-ideal"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "L"; end; theorem :: IDEAL_1:7 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")))) ; theorem :: IDEAL_1:8 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#m1_subset_1 :::"LeftIdeal":::) "of" (Set (Var "L")))) ; theorem :: IDEAL_1:9 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) "is" ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Var "L")))) ; theorem :: IDEAL_1:10 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")))) ; theorem :: IDEAL_1:11 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "is" ($#m1_subset_1 :::"LeftIdeal":::) "of" (Set (Var "L")))) ; theorem :: IDEAL_1:12 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "is" ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Var "L")))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#m1_subset_1 :::"Ideal":::) "of" (Set (Const "R")); :: original: :::"trivial"::: redefine attr "I" is :::"trivial"::: means :: IDEAL_1:def 4 (Bool "I" ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "R" ")" ) ($#k6_domain_1 :::"}"::: ) )); end; :: deftheorem defines :::"trivial"::: IDEAL_1:def 4 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_zfmisc_1 :::"trivial"::: ) ) "iff" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" ) ($#k6_domain_1 :::"}"::: ) )) ")" ))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v7_struct_0 "non" ($#v7_struct_0 :::"trivial"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_subset_1 :::"proper"::: ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R")); end; theorem :: IDEAL_1:13 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "I")))))) ; theorem :: IDEAL_1:14 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "I")))))) ; theorem :: IDEAL_1:15 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"LeftIdeal":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "I")))))) ; theorem :: IDEAL_1:16 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "holds" (Bool (Set (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y"))) ($#r2_hidden :::"in"::: ) (Set (Var "I")))))) ; theorem :: IDEAL_1:17 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "n")) ($#k5_binom :::"*"::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set (Var "I"))))))) ; theorem :: IDEAL_1:18 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "I")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "a")) ($#k2_binom :::"|^"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "I"))))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); func :::"add|"::: "(" "I" "," "R" ")" -> ($#m1_subset_1 :::"BinOp":::) "of" "I" equals :: IDEAL_1:def 5 (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "R") ($#k1_realset1 :::"||"::: ) "I"); end; :: deftheorem defines :::"add|"::: IDEAL_1:def 5 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k1_ideal_1 :::"add|"::: ) "(" (Set (Var "I")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "R"))) ($#k1_realset1 :::"||"::: ) (Set (Var "I")))))); definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); func :::"mult|"::: "(" "I" "," "R" ")" -> ($#m1_subset_1 :::"BinOp":::) "of" "I" equals :: IDEAL_1:def 6 (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" "R") ($#k1_realset1 :::"||"::: ) "I"); end; :: deftheorem defines :::"mult|"::: IDEAL_1:def 6 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k2_ideal_1 :::"mult|"::: ) "(" (Set (Var "I")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "R"))) ($#k1_realset1 :::"||"::: ) (Set (Var "I")))))); definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); func :::"Gr"::: "(" "I" "," "R" ")" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) equals :: IDEAL_1:def 7 (Set ($#g2_algstr_0 :::"addLoopStr"::: ) "(#" "I" "," (Set "(" ($#k1_ideal_1 :::"add|"::: ) "(" "I" "," "R" ")" ")" ) "," (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Set "(" ($#k4_struct_0 :::"0."::: ) "R" ")" ) "," "I" ")" ")" ) "#)" ); end; :: deftheorem defines :::"Gr"::: IDEAL_1:def 7 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k3_ideal_1 :::"Gr"::: ) "(" (Set (Var "I")) "," (Set (Var "R")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#g2_algstr_0 :::"addLoopStr"::: ) "(#" (Set (Var "I")) "," (Set "(" ($#k1_ideal_1 :::"add|"::: ) "(" (Set (Var "I")) "," (Set (Var "R")) ")" ")" ) "," (Set "(" ($#k1_funct_7 :::"In"::: ) "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" ) "," (Set (Var "I")) ")" ")" ) "#)" )))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set ($#k3_ideal_1 :::"Gr"::: ) "(" "I" "," "R" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set ($#k3_ideal_1 :::"Gr"::: ) "(" "I" "," "R" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set ($#k3_ideal_1 :::"Gr"::: ) "(" "I" "," "R" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set ($#k3_ideal_1 :::"Gr"::: ) "(" "I" "," "R" ")" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ; end; theorem :: IDEAL_1:19 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_subset_1 :::"proper"::: ) ) "iff" (Bool (Bool "not" (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "R"))) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) ")" ))) ; theorem :: IDEAL_1:20 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_subset_1 :::"proper"::: ) ) "iff" (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "u")) "is" ($#v1_gcd_1 :::"unital"::: ) )) "holds" (Bool "not" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))))) ")" ))) ; theorem :: IDEAL_1:21 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v1_subset_1 :::"proper"::: ) ) "iff" (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "u")) "is" ($#v1_gcd_1 :::"unital"::: ) )) "holds" (Bool "not" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))))) ")" ))) ; theorem :: IDEAL_1:22 (Bool "for" (Set (Var "R")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l6_algstr_0 :::"comRing":::) "holds" (Bool "(" (Bool (Set (Var "R")) "is" ($#l6_algstr_0 :::"Field":::)) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" ) ($#k6_domain_1 :::"}"::: ) )) "or" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R")))) ")" )) ")" )) ; begin definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); mode :::"LinearCombination"::: "of" "A" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") means :: IDEAL_1:def 8 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool "ex" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "R"(Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" "A" "st" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "u")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "v"))))))); mode :::"LeftLinearCombination"::: "of" "A" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") means :: IDEAL_1:def 9 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" "R"(Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" "A" "st" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a"))))))); mode :::"RightLinearCombination"::: "of" "A" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") means :: IDEAL_1:def 10 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" "R"(Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" "A" "st" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "u"))))))); end; :: deftheorem defines :::"LinearCombination"::: IDEAL_1:def 8 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool "ex" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R"))(Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "A")) "st" (Bool (Set (Set (Var "b3")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "u")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a")) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set (Var "v"))))))) ")" )))); :: deftheorem defines :::"LeftLinearCombination"::: IDEAL_1:def 9 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R"))(Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "A")) "st" (Bool (Set (Set (Var "b3")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "a"))))))) ")" )))); :: deftheorem defines :::"RightLinearCombination"::: IDEAL_1:def 10 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A"))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool "ex" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R"))(Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "A")) "st" (Bool (Set (Set (Var "b3")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "u"))))))) ")" )))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m1_ideal_1 :::"LinearCombination"::: ) "of" "A"; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" "A"; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_finset_1 :::"finite"::: ) ($#v1_finseq_1 :::"FinSequence-like"::: ) ($#v2_finseq_1 :::"FinSubsequence-like"::: ) for ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" "A"; end; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A", "B" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "F" be ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Const "A")); let "G" be ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Const "B")); :: original: :::"^"::: redefine func "F" :::"^"::: "G" -> ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set "A" ($#k4_subset_1 :::"\/"::: ) "B"); end; theorem :: IDEAL_1:23 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "F"))) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:24 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "F")) ($#k1_polynom1 :::"*"::: ) (Set (Var "a"))) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:25 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "f")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Var "f")) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A", "B" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "F" be ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Const "A")); let "G" be ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Const "B")); :: original: :::"^"::: redefine func "F" :::"^"::: "G" -> ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set "A" ($#k4_subset_1 :::"\/"::: ) "B"); end; theorem :: IDEAL_1:26 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "F"))) "is" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:27 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "F")) ($#k1_polynom1 :::"*"::: ) (Set (Var "a"))) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:28 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "f")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Var "f")) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A", "B" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "F" be ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Const "A")); let "G" be ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Const "B")); :: original: :::"^"::: redefine func "F" :::"^"::: "G" -> ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set "A" ($#k4_subset_1 :::"\/"::: ) "B"); end; theorem :: IDEAL_1:29 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "F")) ($#k1_polynom1 :::"*"::: ) (Set (Var "a"))) "is" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:30 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "F")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool (Set (Set (Var "a")) ($#k9_fvsum_1 :::"*"::: ) (Set (Var "F"))) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:31 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "f")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A"))) & (Bool (Set (Var "f")) "is" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A"))) ")" )))) ; theorem :: IDEAL_1:32 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "lc")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "F")) (Bool "ex" (Set (Var "p")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "F"))(Bool "ex" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "lc")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "p")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "e")) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "e")) ($#k12_finseq_1 :::"*>"::: ) ) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "F"))) ")" )))))) ; theorem :: IDEAL_1:33 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "lc")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "F")) (Bool "ex" (Set (Var "p")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "F"))(Bool "ex" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "lc")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "p")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "e")) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "e")) ($#k12_finseq_1 :::"*>"::: ) ) "is" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "F"))) ")" )))))) ; theorem :: IDEAL_1:34 (Bool "for" (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "lc")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "F")) (Bool "ex" (Set (Var "p")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "F"))(Bool "ex" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "S")) "st" (Bool "(" (Bool (Set (Var "lc")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "p")) ($#k8_finseq_1 :::"^"::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "e")) ($#k12_finseq_1 :::"*>"::: ) ))) & (Bool (Set ($#k12_finseq_1 :::"<*"::: ) (Set (Var "e")) ($#k12_finseq_1 :::"*>"::: ) ) "is" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "F"))) ")" )))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "L" be ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Const "A")); let "E" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "R"))) ($#k3_zfmisc_1 :::":]"::: ) ); pred "E" :::"represents"::: "L" means :: IDEAL_1:def 11 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "E") ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "L")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "L"))) "holds" (Bool "(" (Bool (Set "L" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k1_mcart_1 :::"`1_3"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_mcart_1 :::"`2_3"::: ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_mcart_1 :::"`3_3"::: ) ")" ))) & (Bool (Set (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_mcart_1 :::"`2_3"::: ) ) ($#r2_hidden :::"in"::: ) "A") ")" ) ")" ) ")" ); end; :: deftheorem defines :::"represents"::: IDEAL_1:def 11 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k3_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "E")) ($#r1_ideal_1 :::"represents"::: ) (Set (Var "L"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "E"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "L")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k1_mcart_1 :::"`1_3"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_mcart_1 :::"`2_3"::: ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_mcart_1 :::"`3_3"::: ) ")" ))) & (Bool (Set (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_mcart_1 :::"`2_3"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" ) ")" ) ")" ))))); theorem :: IDEAL_1:35 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")) (Bool "ex" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k3_zfmisc_1 :::":]"::: ) ) "st" (Bool (Set (Var "E")) ($#r1_ideal_1 :::"represents"::: ) (Set (Var "L"))))))) ; theorem :: IDEAL_1:36 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "lc")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "F")) (Bool "for" (Set (Var "G")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k3_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k3_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Set (Var "P")) ($#k7_relset_1 :::".:"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Set (Var "E")) ($#r1_ideal_1 :::"represents"::: ) (Set (Var "lc")))) "holds" (Bool "ex" (Set (Var "LC")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "lc"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "LC")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "LC"))))) "holds" (Bool (Set (Set (Var "LC")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "P")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k1_mcart_1 :::"`1_3"::: ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "P")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_mcart_1 :::"`2_3"::: ) ")" ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "P")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_mcart_1 :::"`3_3"::: ) ")" ) ")" ))) ")" ) ")" )))))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "L" be ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Const "A")); let "E" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "R"))) ($#k2_zfmisc_1 :::":]"::: ) ); pred "E" :::"represents"::: "L" means :: IDEAL_1:def 12 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "E") ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "L")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "L"))) "holds" (Bool "(" (Bool (Set "L" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ")" ))) & (Bool (Set (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ) ($#r2_hidden :::"in"::: ) "A") ")" ) ")" ) ")" ); end; :: deftheorem defines :::"represents"::: IDEAL_1:def 12 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "E")) ($#r2_ideal_1 :::"represents"::: ) (Set (Var "L"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "E"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "L")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ")" ))) & (Bool (Set (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" ) ")" ) ")" ))))); theorem :: IDEAL_1:37 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A")) (Bool "ex" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Set (Var "E")) ($#r2_ideal_1 :::"represents"::: ) (Set (Var "L"))))))) ; theorem :: IDEAL_1:38 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "lc")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "F")) (Bool "for" (Set (Var "G")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Set (Var "P")) ($#k7_relset_1 :::".:"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Set (Var "E")) ($#r2_ideal_1 :::"represents"::: ) (Set (Var "lc")))) "holds" (Bool "ex" (Set (Var "LC")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "lc"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "LC")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "LC"))))) "holds" (Bool (Set (Set (Var "LC")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "P")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "P")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ))) ")" ) ")" )))))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "L" be ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Const "A")); let "E" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Const "R"))) ($#k2_zfmisc_1 :::":]"::: ) ); pred "E" :::"represents"::: "L" means :: IDEAL_1:def 13 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) "E") ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "L")) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) "L"))) "holds" (Bool "(" (Bool (Set "L" ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ")" ))) & (Bool (Set (Set "(" "E" ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ) ($#r2_hidden :::"in"::: ) "A") ")" ) ")" ) ")" ); end; :: deftheorem defines :::"represents"::: IDEAL_1:def 13 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "E")) ($#r3_ideal_1 :::"represents"::: ) (Set (Var "L"))) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "E"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "L")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "L"))))) "holds" (Bool "(" (Bool (Set (Set (Var "L")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ")" ))) & (Bool (Set (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) ")" ) ")" ) ")" ) ")" ))))); theorem :: IDEAL_1:39 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "L")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A")) (Bool "ex" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Set (Var "E")) ($#r3_ideal_1 :::"represents"::: ) (Set (Var "L"))))))) ; theorem :: IDEAL_1:40 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "lc")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "F")) (Bool "for" (Set (Var "G")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "S")) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S"))) (Bool "for" (Set (Var "E")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ) "st" (Bool (Bool (Set (Set (Var "P")) ($#k7_relset_1 :::".:"::: ) (Set (Var "F"))) ($#r1_tarski :::"c="::: ) (Set (Var "G"))) & (Bool (Set (Var "E")) ($#r3_ideal_1 :::"represents"::: ) (Set (Var "lc")))) "holds" (Bool "ex" (Set (Var "LC")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "G")) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "lc"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "LC")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "LC"))))) "holds" (Bool (Set (Set (Var "LC")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "P")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "P")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set (Var "E")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ))) ")" ) ")" )))))))) ; theorem :: IDEAL_1:41 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "l")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "l")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) "is" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:42 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "l")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "l")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) "is" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "A"))))))) ; theorem :: IDEAL_1:43 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "l")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A")) (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "l")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")) ")" )) "is" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "A"))))))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "F" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); assume (Bool (Bool "not" (Set (Const "F")) "is" ($#v1_xboole_0 :::"empty"::: ) )) ; func "F" :::"-Ideal"::: -> ($#m1_subset_1 :::"Ideal":::) "of" "L" means :: IDEAL_1:def 14 (Bool "(" (Bool "F" ($#r1_tarski :::"c="::: ) it) & (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" "L" "st" (Bool (Bool "F" ($#r1_tarski :::"c="::: ) (Set (Var "I")))) "holds" (Bool it ($#r1_tarski :::"c="::: ) (Set (Var "I"))) ")" ) ")" ); func "F" :::"-LeftIdeal"::: -> ($#m1_subset_1 :::"LeftIdeal":::) "of" "L" means :: IDEAL_1:def 15 (Bool "(" (Bool "F" ($#r1_tarski :::"c="::: ) it) & (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"LeftIdeal":::) "of" "L" "st" (Bool (Bool "F" ($#r1_tarski :::"c="::: ) (Set (Var "I")))) "holds" (Bool it ($#r1_tarski :::"c="::: ) (Set (Var "I"))) ")" ) ")" ); func "F" :::"-RightIdeal"::: -> ($#m1_subset_1 :::"RightIdeal":::) "of" "L" means :: IDEAL_1:def 16 (Bool "(" (Bool "F" ($#r1_tarski :::"c="::: ) it) & (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"RightIdeal":::) "of" "L" "st" (Bool (Bool "F" ($#r1_tarski :::"c="::: ) (Set (Var "I")))) "holds" (Bool it ($#r1_tarski :::"c="::: ) (Set (Var "I"))) ")" ) ")" ); end; :: deftheorem defines :::"-Ideal"::: IDEAL_1:def 14 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "F")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k7_ideal_1 :::"-Ideal"::: ) )) "iff" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "b3"))) & (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "I")))) "holds" (Bool (Set (Var "b3")) ($#r1_tarski :::"c="::: ) (Set (Var "I"))) ")" ) ")" ) ")" )))); :: deftheorem defines :::"-LeftIdeal"::: IDEAL_1:def 15 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "F")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"LeftIdeal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_ideal_1 :::"-LeftIdeal"::: ) )) "iff" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "b3"))) & (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"LeftIdeal":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "I")))) "holds" (Bool (Set (Var "b3")) ($#r1_tarski :::"c="::: ) (Set (Var "I"))) ")" ) ")" ) ")" )))); :: deftheorem defines :::"-RightIdeal"::: IDEAL_1:def 16 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Bool "not" (Set (Var "F")) "is" ($#v1_xboole_0 :::"empty"::: ) ))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k9_ideal_1 :::"-RightIdeal"::: ) )) "iff" (Bool "(" (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "b3"))) & (Bool "(" "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "F")) ($#r1_tarski :::"c="::: ) (Set (Var "I")))) "holds" (Bool (Set (Var "b3")) ($#r1_tarski :::"c="::: ) (Set (Var "I"))) ")" ) ")" ) ")" )))); theorem :: IDEAL_1:44 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "I")) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "I"))))) ; theorem :: IDEAL_1:45 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"LeftIdeal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "I")) ($#k8_ideal_1 :::"-LeftIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "I"))))) ; theorem :: IDEAL_1:46 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "I")) ($#k9_ideal_1 :::"-RightIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "I"))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#m1_subset_1 :::"Ideal":::) "of" (Set (Const "L")); mode :::"Basis"::: "of" "I" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" "L" means :: IDEAL_1:def 17 (Bool (Set it ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) "I"); end; :: deftheorem defines :::"Basis"::: IDEAL_1:def 17 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "b3")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m4_ideal_1 :::"Basis"::: ) "of" (Set (Var "I"))) "iff" (Bool (Set (Set (Var "b3")) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Var "I"))) ")" )))); theorem :: IDEAL_1:47 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) ; theorem :: IDEAL_1:48 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) ; theorem :: IDEAL_1:49 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k8_ideal_1 :::"-LeftIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) ; theorem :: IDEAL_1:50 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k9_ideal_1 :::"-RightIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) ; theorem :: IDEAL_1:51 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) ; theorem :: IDEAL_1:52 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k8_ideal_1 :::"-LeftIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) ; theorem :: IDEAL_1:53 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k9_ideal_1 :::"-RightIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) ; theorem :: IDEAL_1:54 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "L")) ")" ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) ; theorem :: IDEAL_1:55 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "L")) ")" ) ($#k8_ideal_1 :::"-LeftIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) ; theorem :: IDEAL_1:56 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set "(" ($#k2_struct_0 :::"[#]"::: ) (Set (Var "L")) ")" ) ($#k9_ideal_1 :::"-RightIdeal"::: ) ) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))))) ; theorem :: IDEAL_1:57 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "A")) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k7_ideal_1 :::"-Ideal"::: ) )))) ; theorem :: IDEAL_1:58 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "A")) ($#k8_ideal_1 :::"-LeftIdeal"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k8_ideal_1 :::"-LeftIdeal"::: ) )))) ; theorem :: IDEAL_1:59 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "A")) ($#r1_tarski :::"c="::: ) (Set (Var "B")))) "holds" (Bool (Set (Set (Var "A")) ($#k9_ideal_1 :::"-RightIdeal"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "B")) ($#k9_ideal_1 :::"-RightIdeal"::: ) )))) ; theorem :: IDEAL_1:60 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "F")) ($#k7_ideal_1 :::"-Ideal"::: ) )) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_ideal_1 :::"LinearCombination"::: ) "of" (Set (Var "F")) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "f"))))) ")" )))) ; theorem :: IDEAL_1:61 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "F")) ($#k8_ideal_1 :::"-LeftIdeal"::: ) )) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m2_ideal_1 :::"LeftLinearCombination"::: ) "of" (Set (Var "F")) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "f"))))) ")" )))) ; theorem :: IDEAL_1:62 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "F")) ($#k9_ideal_1 :::"-RightIdeal"::: ) )) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m3_ideal_1 :::"RightLinearCombination"::: ) "of" (Set (Var "F")) "st" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "f"))))) ")" )))) ; theorem :: IDEAL_1:63 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Set (Var "F")) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_ideal_1 :::"-LeftIdeal"::: ) )) & (Bool (Set (Set (Var "F")) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k9_ideal_1 :::"-RightIdeal"::: ) )) ")" ))) ; theorem :: IDEAL_1:64 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "r")) ")" ) where r "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) : (Bool verum) "}" ))) ; theorem :: IDEAL_1:65 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k7_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "r")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "s")) ")" ) ")" ) where r, s "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) : (Bool verum) "}" ))) ; theorem :: IDEAL_1:66 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) )))) ; theorem :: IDEAL_1:67 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "A")) ($#k7_ideal_1 :::"-Ideal"::: ) ))) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set (Var "A")) ($#k7_ideal_1 :::"-Ideal"::: ) ))))) ; theorem :: IDEAL_1:68 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k7_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) )) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k7_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) )) ")" ))) ; theorem :: IDEAL_1:69 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k7_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) )) & (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_tarski :::"c="::: ) (Set (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k7_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) )) ")" ))) ; begin definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) ; let "I" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "R")); func "a" :::"*"::: "I" -> ($#m1_subset_1 :::"Subset":::) "of" "R" equals :: IDEAL_1:def 18 "{" (Set "(" "a" ($#k6_algstr_0 :::"*"::: ) (Set (Var "i")) ")" ) where i "is" ($#m1_subset_1 :::"Element":::) "of" "R" : (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) "I") "}" ; end; :: deftheorem defines :::"*"::: IDEAL_1:def 18 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "a")) ($#k10_ideal_1 :::"*"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "i")) ")" ) where i "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) : (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) "}" )))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "R")); cluster (Set "a" ($#k10_ideal_1 :::"*"::: ) "I") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "R")); cluster (Set "a" ($#k10_ideal_1 :::"*"::: ) "I") -> ($#v1_ideal_1 :::"add-closed"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); let "a" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "R")); cluster (Set "a" ($#k10_ideal_1 :::"*"::: ) "I") -> ($#v3_ideal_1 :::"right-ideal"::: ) ; end; theorem :: IDEAL_1:70 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" ) ($#k10_ideal_1 :::"*"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: IDEAL_1:71 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "R")) ")" ) ($#k10_ideal_1 :::"*"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Var "I"))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); func "I" :::"+"::: "J" -> ($#m1_subset_1 :::"Subset":::) "of" "R" equals :: IDEAL_1:def 19 "{" (Set "(" (Set (Var "a")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "b")) ")" ) where a, b "is" ($#m1_subset_1 :::"Element":::) "of" "R" : (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "I") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "J") ")" ) "}" ; end; :: deftheorem defines :::"+"::: IDEAL_1:def 19 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) "{" (Set "(" (Set (Var "a")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "b")) ")" ) where a, b "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) : (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "J"))) ")" ) "}" ))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I", "J" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k11_ideal_1 :::"+"::: ) "J") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); :: original: :::"+"::: redefine func "I" :::"+"::: "J" -> ($#m1_subset_1 :::"Subset":::) "of" "R"; commutativity (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")) "holds" (Bool (Set (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "J")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "I"))))) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I", "J" be ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k11_ideal_1 :::"+"::: ) "J") -> ($#v1_ideal_1 :::"add-closed"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k11_ideal_1 :::"+"::: ) "J") -> ($#v3_ideal_1 :::"right-ideal"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k11_ideal_1 :::"+"::: ) "J") -> ($#v2_ideal_1 :::"left-ideal"::: ) ; end; theorem :: IDEAL_1:72 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "," (Set (Var "K")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set "(" (Set (Var "J")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J")) ")" ) ($#k11_ideal_1 :::"+"::: ) (Set (Var "K")))))) ; theorem :: IDEAL_1:73 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J")))))) ; theorem :: IDEAL_1:74 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Var "J")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J")))))) ; theorem :: IDEAL_1:75 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "K")) "being" ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Var "K"))) & (Bool (Set (Var "J")) ($#r1_tarski :::"c="::: ) (Set (Var "K")))) "holds" (Bool (Set (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J"))) ($#r1_tarski :::"c="::: ) (Set (Var "K")))))) ; theorem :: IDEAL_1:76 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set ($#k7_domain_1 :::"{"::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k7_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "a")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ")" ) ($#k12_ideal_1 :::"+"::: ) (Set "(" (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "b")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ")" ))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); :: original: :::"/\"::: redefine func "I" :::"/\"::: "J" -> ($#m1_subset_1 :::"Subset":::) "of" "R" equals :: IDEAL_1:def 20 "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" "R" : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "I") & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "J") ")" ) "}" ; end; :: deftheorem defines :::"/\"::: IDEAL_1:def 20 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) : (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "J"))) ")" ) "}" ))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k3_xboole_0 :::"/\"::: ) "J") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I", "J" be ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k3_xboole_0 :::"/\"::: ) "J") -> ($#v1_ideal_1 :::"add-closed"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "R"; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "I", "J" be ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k3_xboole_0 :::"/\"::: ) "J") -> ($#v2_ideal_1 :::"left-ideal"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "R"; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) ; let "I", "J" be ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k3_xboole_0 :::"/\"::: ) "J") -> ($#v3_ideal_1 :::"right-ideal"::: ) for ($#m1_subset_1 :::"Subset":::) "of" "R"; end; theorem :: IDEAL_1:77 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "J")) ($#r1_tarski :::"c="::: ) (Set (Var "I")))) "holds" (Bool (Set (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set "(" (Set (Var "J")) ($#k12_ideal_1 :::"+"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "J")) ($#k12_ideal_1 :::"+"::: ) (Set "(" (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "K")) ")" ))))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); func "I" :::"*'"::: "J" -> ($#m1_subset_1 :::"Subset":::) "of" "R" equals :: IDEAL_1:def 21 "{" (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "s")) ")" ) where s "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") : (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))))) "holds" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "R" "st" (Bool "(" (Bool (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "I") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "J") ")" ))) "}" ; end; :: deftheorem defines :::"*'"::: IDEAL_1:def 21 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) "{" (Set "(" ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "s")) ")" ) where s "is" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) : (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "s"))))) "holds" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Set (Var "s")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")))) & (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "J"))) ")" ))) "}" ))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k14_ideal_1 :::"*'"::: ) "J") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); :: original: :::"*'"::: redefine func "I" :::"*'"::: "J" -> ($#m1_subset_1 :::"Subset":::) "of" "R"; commutativity (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")) "holds" (Bool (Set (Set (Var "I")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "J")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "I"))))) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k14_ideal_1 :::"*'"::: ) "J") -> ($#v1_ideal_1 :::"add-closed"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k14_ideal_1 :::"*'"::: ) "J") -> ($#v3_ideal_1 :::"right-ideal"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k14_ideal_1 :::"*'"::: ) "J") -> ($#v2_ideal_1 :::"left-ideal"::: ) ; end; theorem :: IDEAL_1:78 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" ) ($#k6_domain_1 :::"}"::: ) ) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" ) ($#k6_domain_1 :::"}"::: ) )))) ; theorem :: IDEAL_1:79 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "J"))) ($#r1_tarski :::"c="::: ) (Set (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J"))))))) ; theorem :: IDEAL_1:80 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "," (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k14_ideal_1 :::"*'"::: ) (Set "(" (Set (Var "J")) ($#k12_ideal_1 :::"+"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "J")) ")" ) ($#k12_ideal_1 :::"+"::: ) (Set "(" (Set (Var "I")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "K")) ")" ))))) ; theorem :: IDEAL_1:81 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" (Set (Var "I")) ($#k12_ideal_1 :::"+"::: ) (Set (Var "J")) ")" ) ($#k15_ideal_1 :::"*'"::: ) (Set "(" (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "I")) ($#k15_ideal_1 :::"*'"::: ) (Set (Var "J")))))) ; theorem :: IDEAL_1:82 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J")) ")" ) ($#k14_ideal_1 :::"*'"::: ) (Set "(" (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J")))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); pred "I" "," "J" :::"are_co-prime"::: means :: IDEAL_1:def 22 (Bool (Set "I" ($#k11_ideal_1 :::"+"::: ) "J") ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R")); end; :: deftheorem defines :::"are_co-prime"::: IDEAL_1:def 22 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "I")) "," (Set (Var "J")) ($#r4_ideal_1 :::"are_co-prime"::: ) ) "iff" (Bool (Set (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R")))) ")" ))); theorem :: IDEAL_1:83 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "I")) "," (Set (Var "J")) ($#r4_ideal_1 :::"are_co-prime"::: ) )) "holds" (Bool (Set (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J"))) ($#r1_tarski :::"c="::: ) (Set (Set "(" (Set (Var "I")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "J")) ")" ) ($#k14_ideal_1 :::"*'"::: ) (Set "(" (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J")) ")" ))))) ; theorem :: IDEAL_1:84 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "I")) "," (Set (Var "J")) ($#r4_ideal_1 :::"are_co-prime"::: ) )) "holds" (Bool (Set (Set (Var "I")) ($#k15_ideal_1 :::"*'"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J"))))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) ; let "I", "J" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); func "I" :::"%"::: "J" -> ($#m1_subset_1 :::"Subset":::) "of" "R" equals :: IDEAL_1:def 23 "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element":::) "of" "R" : (Bool (Set (Set (Var "a")) ($#k10_ideal_1 :::"*"::: ) "J") ($#r1_tarski :::"c="::: ) "I") "}" ; end; :: deftheorem defines :::"%"::: IDEAL_1:def 23 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "J"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) : (Bool (Set (Set (Var "a")) ($#k10_ideal_1 :::"*"::: ) (Set (Var "J"))) ($#r1_tarski :::"c="::: ) (Set (Var "I"))) "}" ))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k16_ideal_1 :::"%"::: ) "J") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k16_ideal_1 :::"%"::: ) "J") -> ($#v1_ideal_1 :::"add-closed"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I", "J" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set "I" ($#k16_ideal_1 :::"%"::: ) "J") -> ($#v2_ideal_1 :::"left-ideal"::: ) ; cluster (Set "I" ($#k16_ideal_1 :::"%"::: ) "J") -> ($#v3_ideal_1 :::"right-ideal"::: ) ; end; theorem :: IDEAL_1:85 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "J"))))))) ; theorem :: IDEAL_1:86 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "J")) ")" ) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "J"))) ($#r1_tarski :::"c="::: ) (Set (Var "I")))))) ; theorem :: IDEAL_1:87 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "J")) ")" ) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "J"))) ($#r1_tarski :::"c="::: ) (Set (Var "I")))))) ; theorem :: IDEAL_1:88 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "," (Set (Var "K")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "J")) ")" ) ($#k16_ideal_1 :::"%"::: ) (Set (Var "K"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set "(" (Set (Var "J")) ($#k15_ideal_1 :::"*'"::: ) (Set (Var "K")) ")" )))))) ; theorem :: IDEAL_1:89 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l4_algstr_0 :::"multLoopStr"::: ) (Bool "for" (Set (Var "I")) "," (Set (Var "J")) "," (Set (Var "K")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set "(" (Set (Var "J")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "K")) ")" ) ($#k16_ideal_1 :::"%"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "J")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "I")) ")" ) ($#k13_ideal_1 :::"/\"::: ) (Set "(" (Set (Var "K")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "I")) ")" ))))) ; theorem :: IDEAL_1:90 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "," (Set (Var "K")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set "(" (Set (Var "J")) ($#k11_ideal_1 :::"+"::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "J")) ")" ) ($#k13_ideal_1 :::"/\"::: ) (Set "(" (Set (Var "I")) ($#k16_ideal_1 :::"%"::: ) (Set (Var "K")) ")" )))))) ; definitionlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); func :::"sqrt"::: "I" -> ($#m1_subset_1 :::"Subset":::) "of" "R" equals :: IDEAL_1:def 24 "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element":::) "of" "R" : (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "a")) ($#k2_binom :::"|^"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) "I")) "}" ; end; :: deftheorem defines :::"sqrt"::: IDEAL_1:def 24 : (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k17_ideal_1 :::"sqrt"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "a")) where a "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) : (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "a")) ($#k2_binom :::"|^"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set (Var "I")))) "}" ))); registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set ($#k17_ideal_1 :::"sqrt"::: ) "I") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set ($#k17_ideal_1 :::"sqrt"::: ) "I") -> ($#v1_ideal_1 :::"add-closed"::: ) ; end; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "R")); cluster (Set ($#k17_ideal_1 :::"sqrt"::: ) "I") -> ($#v2_ideal_1 :::"left-ideal"::: ) ; cluster (Set ($#k17_ideal_1 :::"sqrt"::: ) "I") -> ($#v3_ideal_1 :::"right-ideal"::: ) ; end; theorem :: IDEAL_1:91 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k17_ideal_1 :::"sqrt"::: ) (Set (Var "I")))) "iff" (Bool "ex" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "a")) ($#k2_binom :::"|^"::: ) (Set (Var "n"))) ($#r2_hidden :::"in"::: ) (Set ($#k17_ideal_1 :::"sqrt"::: ) (Set (Var "I"))))) ")" )))) ; theorem :: IDEAL_1:92 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "J")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set ($#k17_ideal_1 :::"sqrt"::: ) (Set "(" (Set (Var "I")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "J")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k17_ideal_1 :::"sqrt"::: ) (Set "(" (Set (Var "I")) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "J")) ")" )))))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#m1_subset_1 :::"Ideal":::) "of" (Set (Const "L")); attr "I" is :::"finitely_generated"::: means :: IDEAL_1:def 25 (Bool "ex" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" "L" "st" (Bool "I" ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k7_ideal_1 :::"-Ideal"::: ) ))); end; :: deftheorem defines :::"finitely_generated"::: IDEAL_1:def 25 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v5_ideal_1 :::"finitely_generated"::: ) ) "iff" (Bool "ex" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k7_ideal_1 :::"-Ideal"::: ) ))) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ($#v5_ideal_1 :::"finitely_generated"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L")); end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "F" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "L")); cluster (Set "F" ($#k7_ideal_1 :::"-Ideal"::: ) ) -> ($#v5_ideal_1 :::"finitely_generated"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; attr "L" is :::"Noetherian"::: means :: IDEAL_1:def 26 (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" "L" "holds" (Bool (Set (Var "I")) "is" ($#v5_ideal_1 :::"finitely_generated"::: ) )); end; :: deftheorem defines :::"Noetherian"::: IDEAL_1:def 26 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v6_ideal_1 :::"Noetherian"::: ) ) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "I")) "is" ($#v5_ideal_1 :::"finitely_generated"::: ) )) ")" )); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_int_3 :::"Euclidian"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "I" be ($#m1_subset_1 :::"Ideal":::) "of" (Set (Const "L")); attr "I" is :::"principal"::: means :: IDEAL_1:def 27 (Bool "ex" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element":::) "of" "L" "st" (Bool "I" ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "e")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ))); end; :: deftheorem defines :::"principal"::: IDEAL_1:def 27 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "I")) "is" ($#v7_ideal_1 :::"principal"::: ) ) "iff" (Bool "ex" (Set (Var "e")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "e")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) ))) ")" ))); definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; attr "L" is :::"PID"::: means :: IDEAL_1:def 28 (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" "L" "holds" (Bool (Set (Var "I")) "is" ($#v7_ideal_1 :::"principal"::: ) )); end; :: deftheorem defines :::"PID"::: IDEAL_1:def 28 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "L")) "is" ($#v8_ideal_1 :::"PID"::: ) ) "iff" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "L")) "holds" (Bool (Set (Var "I")) "is" ($#v7_ideal_1 :::"principal"::: ) )) ")" )); theorem :: IDEAL_1:93 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "F")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" ) ($#k6_domain_1 :::"}"::: ) ))) "holds" (Bool "ex" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "L")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "L")))) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "F"))) ")" )))) ; theorem :: IDEAL_1:94 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#v1_int_3 :::"Euclidian"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Var "R")) "is" ($#v8_ideal_1 :::"PID"::: ) )) ; theorem :: IDEAL_1:95 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "st" (Bool (Bool (Set (Var "L")) "is" ($#v8_ideal_1 :::"PID"::: ) )) "holds" (Bool (Set (Var "L")) "is" ($#v6_ideal_1 :::"Noetherian"::: ) )) ; registration cluster (Set ($#k1_int_3 :::"INT.Ring"::: ) ) -> ($#v6_ideal_1 :::"Noetherian"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v6_ideal_1 :::"Noetherian"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; theorem :: IDEAL_1:96 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_algstr_1 :::"left_zeroed"::: ) ($#v6_ideal_1 :::"Noetherian"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "ex" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Set (Var "C")) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k7_ideal_1 :::"-Ideal"::: ) )) ")" )))) ; theorem :: IDEAL_1:97 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "B")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) (Bool "ex" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "st" (Bool "(" (Bool (Set (Var "C")) ($#r1_tarski :::"c="::: ) (Set (Var "B"))) & (Bool (Set (Set (Var "C")) ($#k7_ideal_1 :::"-Ideal"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "B")) ($#k7_ideal_1 :::"-Ideal"::: ) )) ")" )) ")" )) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "R")) (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "a")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "a")) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k7_ideal_1 :::"-Ideal"::: ) ))))) ; registrationlet "X", "Y" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "X")) "," (Set (Const "Y")); let "A" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "X")); cluster (Set "f" ($#k5_relat_1 :::"|"::: ) "A") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: IDEAL_1:98 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "R")) (Bool "ex" (Set (Var "m")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "a")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r2_hidden :::"in"::: ) (Set (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "a")) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "m")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k7_ideal_1 :::"-Ideal"::: ) ))) ")" )) "holds" (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ")" ) "holds" (Bool "(" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) "is" (Bool "not" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "R"))))) "or" (Bool "ex" (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Bool "not" (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) ($#r2_xboole_0 :::"c<"::: ) (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "k"))))) ")" )) ")" ))) ; theorem :: IDEAL_1:99 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "st" (Bool (Bool "(" "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "(" ($#k9_setfam_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ")" ) "holds" (Bool "(" (Bool "ex" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "i"))) "is" (Bool "not" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "R"))))) "or" (Bool "ex" (Set (Var "j")) "," (Set (Var "k")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k"))) & (Bool (Bool "not" (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "j"))) ($#r2_xboole_0 :::"c<"::: ) (Set (Set (Var "F")) ($#k3_funct_2 :::"."::: ) (Set (Var "k"))))) ")" )) ")" ) ")" )) "holds" (Bool (Set (Var "R")) "is" ($#v6_ideal_1 :::"Noetherian"::: ) )) ;