:: QUOFIELD semantic presentation begin definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) ; func :::"Q."::: "I" -> ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "I") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "I") ($#k2_zfmisc_1 :::":]"::: ) ) means :: QUOFIELD:def 1 (Bool "for" (Set (Var "u")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "I" "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_domain_1 :::"]"::: ) )) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "I")) ")" )) ")" )); end; :: deftheorem defines :::"Q."::: QUOFIELD:def 1 : (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "I"))) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "I"))) ($#k2_zfmisc_1 :::":]"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "u")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "I")) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "a")) "," (Set (Var "b")) ($#k1_domain_1 :::"]"::: ) )) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "I")))) ")" )) ")" )) ")" ))); theorem :: QUOFIELD:1 (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) "holds" (Bool (Bool "not" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; registrationlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; cluster (Set ($#k1_quofield :::"Q."::: ) "I") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: QUOFIELD:2 (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set (Var "u")) ($#k3_domain_1 :::"`2"::: ) ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "I")))))) ; definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "u", "v" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))); func :::"padd"::: "(" "u" "," "v" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") equals :: QUOFIELD:def 2 (Set ($#k1_domain_1 :::"["::: ) (Set "(" (Set "(" (Set "(" "u" ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "v" ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" "v" ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "u" ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" "u" ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "v" ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k1_domain_1 :::"]"::: ) ); end; :: deftheorem defines :::"padd"::: QUOFIELD:def 2 : (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k2_quofield :::"padd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set "(" (Set "(" (Set "(" (Set (Var "u")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set "(" (Set (Var "v")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "u")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ")" ) "," (Set "(" (Set "(" (Set (Var "u")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k1_domain_1 :::"]"::: ) )))); definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "u", "v" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))); func :::"pmult"::: "(" "u" "," "v" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") equals :: QUOFIELD:def 3 (Set ($#k1_domain_1 :::"["::: ) (Set "(" (Set "(" "u" ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "v" ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ) "," (Set "(" (Set "(" "u" ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "v" ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k1_domain_1 :::"]"::: ) ); end; :: deftheorem defines :::"pmult"::: QUOFIELD:def 3 : (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k3_quofield :::"pmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set "(" (Set "(" (Set (Var "u")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ) "," (Set "(" (Set "(" (Set (Var "u")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k1_domain_1 :::"]"::: ) )))); theorem :: QUOFIELD:3 (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k2_quofield :::"padd"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k2_quofield :::"padd"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_quofield :::"padd"::: ) "(" (Set "(" ($#k2_quofield :::"padd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" )))) ; theorem :: QUOFIELD:4 (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k3_quofield :::"pmult"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k3_quofield :::"pmult"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_quofield :::"pmult"::: ) "(" (Set "(" ($#k3_quofield :::"pmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" )))) ; definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "u", "v" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))); :: original: :::"padd"::: redefine func :::"padd"::: "(" "u" "," "v" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I"); commutativity (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))) "holds" (Bool (Set ($#k2_quofield :::"padd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_quofield :::"padd"::: ) "(" (Set (Var "v")) "," (Set (Var "u")) ")" ))) ; end; definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "u", "v" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))); :: original: :::"pmult"::: redefine func :::"pmult"::: "(" "u" "," "v" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I"); commutativity (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))) "holds" (Bool (Set ($#k3_quofield :::"pmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_quofield :::"pmult"::: ) "(" (Set (Var "v")) "," (Set (Var "u")) ")" ))) ; end; definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; let "u" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))); func :::"QClass."::: "u" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_quofield :::"Q."::: ) "I" ")" ) means :: QUOFIELD:def 4 (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "u" ($#k3_domain_1 :::"`2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "u" ($#k2_domain_1 :::"`1"::: ) ")" ))) ")" )); end; :: deftheorem defines :::"QClass."::: QUOFIELD:def 4 : (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_quofield :::"Q."::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_quofield :::"QClass."::: ) (Set (Var "u")))) "iff" (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "u")) ($#k3_domain_1 :::"`2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "u")) ($#k2_domain_1 :::"`1"::: ) ")" ))) ")" )) ")" )))); theorem :: QUOFIELD:5 (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set ($#k6_quofield :::"QClass."::: ) (Set (Var "u")))))) ; registrationlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; let "u" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))); cluster (Set ($#k6_quofield :::"QClass."::: ) "u") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; func :::"Quot."::: "I" -> ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" ($#k1_quofield :::"Q."::: ) "I" ")" ) means :: QUOFIELD:def 5 (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_quofield :::"Q."::: ) "I" ")" ) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_quofield :::"QClass."::: ) (Set (Var "u"))))) ")" )); end; :: deftheorem defines :::"Quot."::: QUOFIELD:def 5 : (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Subset-Family":::) "of" (Set "(" ($#k1_quofield :::"Q."::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_quofield :::"Q."::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "A")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool "ex" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k6_quofield :::"QClass."::: ) (Set (Var "u"))))) ")" )) ")" ))); theorem :: QUOFIELD:6 (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) "holds" (Bool (Bool "not" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "is" ($#v1_xboole_0 :::"empty"::: ) ))) ; registrationlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; cluster (Set ($#k7_quofield :::"Quot."::: ) "I") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; theorem :: QUOFIELD:7 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool (Bool "ex" (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "st" (Bool "(" (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "w"))) & (Bool (Set (Var "v")) ($#r2_hidden :::"in"::: ) (Set (Var "w"))) ")" ))) "holds" (Bool (Set (Set "(" (Set (Var "u")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k3_domain_1 :::"`2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "u")) ($#k3_domain_1 :::"`2"::: ) ")" ))))) ; theorem :: QUOFIELD:8 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "st" (Bool (Bool (Set (Var "u")) ($#r1_xboole_0 :::"meets"::: ) (Set (Var "v")))) "holds" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set (Var "v"))))) ; begin definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); let "u", "v" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Const "I"))); func :::"qadd"::: "(" "u" "," "v" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") means :: QUOFIELD:def 6 (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "u") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "v") & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"qadd"::: QUOFIELD:def 6 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "b4")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" )) "iff" (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b4"))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "u"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "v"))) & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" (Set (Var "b")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" ) ")" ))) ")" )) ")" )) ")" ))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); let "u", "v" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Const "I"))); func :::"qmult"::: "(" "u" "," "v" ")" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") means :: QUOFIELD:def 7 (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "u") & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) "v") & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"qmult"::: QUOFIELD:def 7 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "b4")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" )) "iff" (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b4"))) "iff" (Bool "ex" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "u"))) & (Bool (Set (Var "b")) ($#r2_hidden :::"in"::: ) (Set (Var "v"))) & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k3_domain_1 :::"`2"::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "b")) ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ))) ")" )) ")" )) ")" ))); definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; let "u" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Const "I"))); :: original: :::"QClass."::: redefine func :::"QClass."::: "u" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I"); end; theorem :: QUOFIELD:9 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k8_quofield :::"qadd"::: ) "(" (Set "(" ($#k10_quofield :::"QClass."::: ) (Set (Var "u")) ")" ) "," (Set "(" ($#k10_quofield :::"QClass."::: ) (Set (Var "v")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_quofield :::"QClass."::: ) (Set "(" ($#k4_quofield :::"padd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ))))) ; theorem :: QUOFIELD:10 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set "(" ($#k10_quofield :::"QClass."::: ) (Set (Var "u")) ")" ) "," (Set "(" ($#k10_quofield :::"QClass."::: ) (Set (Var "v")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k10_quofield :::"QClass."::: ) (Set "(" ($#k5_quofield :::"pmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ))))) ; definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"q0."::: "I" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") means :: QUOFIELD:def 8 (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "I")) ")" )); end; :: deftheorem defines :::"q0."::: QUOFIELD:def 8 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "I")))) ")" )) ")" ))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"q1."::: "I" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") means :: QUOFIELD:def 9 (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) )) ")" )); end; :: deftheorem defines :::"q1."::: QUOFIELD:def 9 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "b2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) )) ")" )) ")" ))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); let "u" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Const "I"))); func :::"qaddinv"::: "u" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") means :: QUOFIELD:def 10 (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "u") & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"qaddinv"::: QUOFIELD:def 10 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "b3")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k13_quofield :::"qaddinv"::: ) (Set (Var "u")))) "iff" (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "u"))) & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" ) ")" ))) ")" )) ")" )) ")" ))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); let "u" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Const "I"))); assume (Bool (Set (Const "u")) ($#r1_hidden :::"<>"::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Const "I")))) ; func :::"qmultinv"::: "u" -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") means :: QUOFIELD:def 11 (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) "u") & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"qmultinv"::: QUOFIELD:def 11 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I"))))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k14_quofield :::"qmultinv"::: ) (Set (Var "u")))) "iff" (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "a")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "u"))) & (Bool (Set (Set "(" (Set (Var "z")) ($#k2_domain_1 :::"`1"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k2_domain_1 :::"`1"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k3_domain_1 :::"`2"::: ) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_domain_1 :::"`2"::: ) ")" ))) ")" )) ")" )) ")" )))); theorem :: QUOFIELD:11 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_quofield :::"qadd"::: ) "(" (Set "(" ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" )) & (Bool (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "v")) "," (Set (Var "u")) ")" )) ")" ))) ; theorem :: QUOFIELD:12 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k11_quofield :::"q0."::: ) (Set (Var "I")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) & (Bool (Set ($#k8_quofield :::"qadd"::: ) "(" (Set "(" ($#k11_quofield :::"q0."::: ) (Set (Var "I")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))) ; theorem :: QUOFIELD:13 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_quofield :::"qmult"::: ) "(" (Set "(" ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" )) & (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "v")) "," (Set (Var "u")) ")" )) ")" ))) ; theorem :: QUOFIELD:14 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k12_quofield :::"q1."::: ) (Set (Var "I")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) & (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set "(" ($#k12_quofield :::"q1."::: ) (Set (Var "I")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))) ; theorem :: QUOFIELD:15 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set "(" ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_quofield :::"qadd"::: ) "(" (Set "(" ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "w")) ")" ")" ) "," (Set "(" ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" )))) ; theorem :: QUOFIELD:16 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_quofield :::"qadd"::: ) "(" (Set "(" ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set "(" ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "w")) ")" ")" ) ")" )))) ; theorem :: QUOFIELD:17 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k13_quofield :::"qaddinv"::: ) (Set (Var "u")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I")))) & (Bool (Set ($#k8_quofield :::"qadd"::: ) "(" (Set "(" ($#k13_quofield :::"qaddinv"::: ) (Set (Var "u")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I")))) ")" ))) ; theorem :: QUOFIELD:18 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I"))))) "holds" (Bool "(" (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set "(" ($#k14_quofield :::"qmultinv"::: ) (Set (Var "u")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I")))) & (Bool (Set ($#k9_quofield :::"qmult"::: ) "(" (Set "(" ($#k14_quofield :::"qmultinv"::: ) (Set (Var "u")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I")))) ")" ))) ; theorem :: QUOFIELD:19 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I"))) ($#r1_hidden :::"<>"::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I"))))) ; definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"quotadd"::: "I" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) "I" ")" ) means :: QUOFIELD:def 12 (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") "holds" (Bool (Set it ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ))); end; :: deftheorem defines :::"quotadd"::: QUOFIELD:def 12 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set (Var "b2")) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k8_quofield :::"qadd"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ))) ")" ))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"quotmult"::: "I" -> ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) "I" ")" ) means :: QUOFIELD:def 13 (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") "holds" (Bool (Set it ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ))); end; :: deftheorem defines :::"quotmult"::: QUOFIELD:def 13 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set (Var "b2")) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k9_quofield :::"qmult"::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ))) ")" ))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"quotaddinv"::: "I" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) "I" ")" ) means :: QUOFIELD:def 14 (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k13_quofield :::"qaddinv"::: ) (Set (Var "u"))))); end; :: deftheorem defines :::"quotaddinv"::: QUOFIELD:def 14 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k17_quofield :::"quotaddinv"::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k13_quofield :::"qaddinv"::: ) (Set (Var "u"))))) ")" ))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"quotmultinv"::: "I" -> ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) "I" ")" ) means :: QUOFIELD:def 15 (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) "I") "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k14_quofield :::"qmultinv"::: ) (Set (Var "u"))))); end; :: deftheorem defines :::"quotmultinv"::: QUOFIELD:def 15 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"UnOp":::) "of" (Set "(" ($#k7_quofield :::"Quot."::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k18_quofield :::"quotmultinv"::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k14_quofield :::"qmultinv"::: ) (Set (Var "u"))))) ")" ))); theorem :: QUOFIELD:20 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set "(" (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" )))) ; theorem :: QUOFIELD:21 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "v")) "," (Set (Var "u")) ")" )))) ; theorem :: QUOFIELD:22 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set "(" ($#k11_quofield :::"q0."::: ) (Set (Var "I")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) & (Bool (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" ($#k11_quofield :::"q0."::: ) (Set (Var "I")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))) ; theorem :: QUOFIELD:23 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set "(" (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" )))) ; theorem :: QUOFIELD:24 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "v")) "," (Set (Var "u")) ")" )))) ; theorem :: QUOFIELD:25 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set "(" ($#k12_quofield :::"q1."::: ) (Set (Var "I")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) & (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" ($#k12_quofield :::"q1."::: ) (Set (Var "I")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "u"))) ")" ))) ; theorem :: QUOFIELD:26 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set (Var "w")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "w")) ")" ")" ) "," (Set "(" (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" )))) ; theorem :: QUOFIELD:27 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set "(" (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ")" ) "," (Set "(" (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "w")) ")" ")" ) ")" )))) ; theorem :: QUOFIELD:28 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set "(" (Set "(" ($#k17_quofield :::"quotaddinv"::: ) (Set (Var "I")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "u")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I")))) & (Bool (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k17_quofield :::"quotaddinv"::: ) (Set (Var "I")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "u")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I")))) ")" ))) ; theorem :: QUOFIELD:29 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I"))) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set "(" (Set "(" ($#k18_quofield :::"quotmultinv"::: ) (Set (Var "I")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "u")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I")))) & (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k5_binop_1 :::"."::: ) "(" (Set "(" (Set "(" ($#k18_quofield :::"quotmultinv"::: ) (Set (Var "I")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "u")) ")" ) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I")))) ")" ))) ; begin definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"the_Field_of_Quotients"::: "I" -> ($#v36_algstr_0 :::"strict"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) equals :: QUOFIELD:def 16 (Set ($#g6_algstr_0 :::"doubleLoopStr"::: ) "(#" (Set "(" ($#k7_quofield :::"Quot."::: ) "I" ")" ) "," (Set "(" ($#k15_quofield :::"quotadd"::: ) "I" ")" ) "," (Set "(" ($#k16_quofield :::"quotmult"::: ) "I" ")" ) "," (Set "(" ($#k12_quofield :::"q1."::: ) "I" ")" ) "," (Set "(" ($#k11_quofield :::"q0."::: ) "I" ")" ) "#)" ); end; :: deftheorem defines :::"the_Field_of_Quotients"::: QUOFIELD:def 16 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I"))) ($#r1_hidden :::"="::: ) (Set ($#g6_algstr_0 :::"doubleLoopStr"::: ) "(#" (Set "(" ($#k7_quofield :::"Quot."::: ) (Set (Var "I")) ")" ) "," (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) "," (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) "," (Set "(" ($#k12_quofield :::"q1."::: ) (Set (Var "I")) ")" ) "," (Set "(" ($#k11_quofield :::"q0."::: ) (Set (Var "I")) ")" ) "#)" ))); registrationlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); cluster (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) "I") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ; end; theorem :: QUOFIELD:30 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_quofield :::"Quot."::: ) (Set (Var "I")))) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r1_funct_2 :::"="::: ) (Set ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")))) & (Bool (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r1_funct_2 :::"="::: ) (Set ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")))) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I")))) & (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I")))) ")" )) ; theorem :: QUOFIELD:31 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )))) ; theorem :: QUOFIELD:32 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set "(" ($#k17_quofield :::"quotaddinv"::: ) (Set (Var "I")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "u"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )))) ; theorem :: QUOFIELD:33 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" ) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )))) ; theorem :: QUOFIELD:34 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set "(" ($#k18_quofield :::"quotmultinv"::: ) (Set (Var "I")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "u"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )))) ; theorem :: QUOFIELD:35 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set (Var "u")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k15_quofield :::"quotadd"::: ) (Set (Var "I")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" )))) ; registrationlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); cluster (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) "I") -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v36_algstr_0 :::"strict"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ; end; theorem :: QUOFIELD:36 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k17_quofield :::"quotaddinv"::: ) (Set (Var "I")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "u")))))) ; theorem :: QUOFIELD:37 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set (Var "u")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k16_quofield :::"quotmult"::: ) (Set (Var "I")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "u")) "," (Set (Var "v")) ")" )))) ; registrationlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); cluster (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) "I") -> ($#v36_algstr_0 :::"strict"::: ) ($#v5_group_1 :::"commutative"::: ) ; end; registrationlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); cluster (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) "I") -> ($#v36_algstr_0 :::"strict"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ; end; theorem :: QUOFIELD:38 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool "(" (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k12_quofield :::"q1."::: ) (Set (Var "I")))) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k11_quofield :::"q0."::: ) (Set (Var "I")))) ")" )) ; theorem :: QUOFIELD:39 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set "(" (Set (Var "u")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k1_algstr_0 :::"+"::: ) (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "w")) ")" ))))) ; theorem :: QUOFIELD:40 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set (Var "u")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "u")))))) ; theorem :: QUOFIELD:41 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set (Var "u")) ($#k1_algstr_0 :::"+"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "u"))))) ; theorem :: QUOFIELD:42 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set "(" ($#k5_struct_0 :::"1."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set (Var "u"))))) ; theorem :: QUOFIELD:43 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set (Var "u")) ($#k8_group_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k8_group_1 :::"*"::: ) (Set (Var "u")))))) ; theorem :: QUOFIELD:44 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "," (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool (Set (Set "(" (Set (Var "u")) ($#k8_group_1 :::"*"::: ) (Set (Var "v")) ")" ) ($#k8_group_1 :::"*"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "u")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k8_group_1 :::"*"::: ) (Set (Var "w")) ")" ))))) ; theorem :: QUOFIELD:45 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "u")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )))) "holds" (Bool "ex" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "st" (Bool (Set (Set (Var "u")) ($#k8_group_1 :::"*"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )))))) ; theorem :: QUOFIELD:46 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I"))) "is" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_group_1 :::"unital"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) )) ; registrationlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); cluster (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) "I") -> ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v36_algstr_0 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ; end; theorem :: QUOFIELD:47 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )))) "holds" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "I")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "I"))))) "holds" (Bool "for" (Set (Var "u")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k10_quofield :::"QClass."::: ) (Set (Var "u")))) & (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "a")) "," (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "I")) ")" ) ($#k1_domain_1 :::"]"::: ) ))) "holds" (Bool "for" (Set (Var "v")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) (Set (Var "I"))) "st" (Bool (Bool (Set (Var "v")) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "I")) ")" ) "," (Set (Var "a")) ($#k1_domain_1 :::"]"::: ) ))) "holds" (Bool (Set (Set (Var "x")) ($#k11_algstr_0 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k10_quofield :::"QClass."::: ) (Set (Var "v"))))))))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) for ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; end; definitionlet "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "F")); func "x" :::"/"::: "y" -> ($#m1_subset_1 :::"Element":::) "of" "F" equals :: QUOFIELD:def 17 (Set "x" ($#k8_group_1 :::"*"::: ) (Set "(" "y" ($#k11_algstr_0 :::"""::: ) ")" )); end; :: deftheorem defines :::"/"::: QUOFIELD:def 17 : (Bool "for" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "holds" (Bool (Set (Set (Var "x")) ($#k20_quofield :::"/"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k11_algstr_0 :::"""::: ) ")" ))))); theorem :: QUOFIELD:48 (Bool "for" (Set (Var "F")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v5_group_1 :::"commutative"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) & (Bool (Set (Var "d")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k20_quofield :::"/"::: ) (Set (Var "b")) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "c")) ($#k20_quofield :::"/"::: ) (Set (Var "d")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "c")) ")" ) ($#k20_quofield :::"/"::: ) (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ))))) ; theorem :: QUOFIELD:49 (Bool "for" (Set (Var "F")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v5_group_1 :::"commutative"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "," (Set (Var "c")) "," (Set (Var "d")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "F")) "st" (Bool (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F")))) & (Bool (Set (Var "d")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set "(" (Set (Var "a")) ($#k20_quofield :::"/"::: ) (Set (Var "b")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "c")) ($#k20_quofield :::"/"::: ) (Set (Var "d")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "a")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "c")) ($#k8_group_1 :::"*"::: ) (Set (Var "b")) ")" ) ")" ) ($#k20_quofield :::"/"::: ) (Set "(" (Set (Var "b")) ($#k8_group_1 :::"*"::: ) (Set (Var "d")) ")" ))))) ; begin definitionlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "R")) "," (Set (Const "S")); attr "f" is :::"RingHomomorphism"::: means :: QUOFIELD:def 18 (Bool "(" (Bool "f" "is" ($#v13_vectsp_1 :::"additive"::: ) ) & (Bool "f" "is" ($#v1_group_6 :::"multiplicative"::: ) ) & (Bool "f" "is" ($#v6_group_1 :::"unity-preserving"::: ) ) ")" ); end; :: deftheorem defines :::"RingHomomorphism"::: QUOFIELD:def 18 : (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#v13_vectsp_1 :::"additive"::: ) ) & (Bool (Set (Var "f")) "is" ($#v1_group_6 :::"multiplicative"::: ) ) & (Bool (Set (Var "f")) "is" ($#v6_group_1 :::"unity-preserving"::: ) ) ")" ) ")" ))); registrationlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v1_quofield :::"RingHomomorphism"::: ) -> ($#v6_group_1 :::"unity-preserving"::: ) ($#v1_group_6 :::"multiplicative"::: ) ($#v13_vectsp_1 :::"additive"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v6_group_1 :::"unity-preserving"::: ) ($#v1_group_6 :::"multiplicative"::: ) ($#v13_vectsp_1 :::"additive"::: ) -> ($#v1_quofield :::"RingHomomorphism"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "R")) "," (Set (Const "S")); attr "f" is :::"RingEpimorphism"::: means :: QUOFIELD:def 19 (Bool "(" (Bool "f" "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) "f") ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S")) ")" ); attr "f" is :::"RingMonomorphism"::: means :: QUOFIELD:def 20 (Bool "(" (Bool "f" "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool "f" "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ); end; :: deftheorem defines :::"RingEpimorphism"::: QUOFIELD:def 19 : (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_quofield :::"RingEpimorphism"::: ) ) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "S")))) ")" ) ")" ))); :: deftheorem defines :::"RingMonomorphism"::: QUOFIELD:def 20 : (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) ) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ) ")" ) ")" ))); notationlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "R")) "," (Set (Const "S")); synonym :::"embedding"::: "f" for :::"RingMonomorphism"::: ; end; definitionlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "R")) "," (Set (Const "S")); attr "f" is :::"RingIsomorphism"::: means :: QUOFIELD:def 21 (Bool "(" (Bool "f" "is" ($#v3_quofield :::"RingMonomorphism"::: ) ) & (Bool "f" "is" ($#v2_quofield :::"RingEpimorphism"::: ) ) ")" ); end; :: deftheorem defines :::"RingIsomorphism"::: QUOFIELD:def 21 : (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v4_quofield :::"RingIsomorphism"::: ) ) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) ) & (Bool (Set (Var "f")) "is" ($#v2_quofield :::"RingEpimorphism"::: ) ) ")" ) ")" ))); registrationlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v4_quofield :::"RingIsomorphism"::: ) -> ($#v2_quofield :::"RingEpimorphism"::: ) ($#v3_quofield :::"RingMonomorphism"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_quofield :::"RingEpimorphism"::: ) ($#v3_quofield :::"RingMonomorphism"::: ) -> ($#v4_quofield :::"RingIsomorphism"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "R") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "S") ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: QUOFIELD:50 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "R")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "S")))))) ; theorem :: QUOFIELD:51 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "S")))) "iff" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R")))) ")" )))) ; theorem :: QUOFIELD:52 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v5_group_1 :::"commutative"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k11_algstr_0 :::"""::: ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k11_algstr_0 :::"""::: ) ))))) ; theorem :: QUOFIELD:53 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v5_group_1 :::"commutative"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "R"))))) "holds" (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "y")) ($#k11_algstr_0 :::"""::: ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" ) ($#k11_algstr_0 :::"""::: ) ")" )))))) ; theorem :: QUOFIELD:54 (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "," (Set (Var "T")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )) "holds" (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "T")) "st" (Bool (Bool (Set (Var "g")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )) "holds" (Bool (Set (Set (Var "g")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )))) ; theorem :: QUOFIELD:55 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set ($#k3_struct_0 :::"id"::: ) (Set (Var "R"))) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )) ; registrationlet "R" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k3_struct_0 :::"id"::: ) "R") -> ($#v1_quofield :::"RingHomomorphism"::: ) ; end; definitionlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; pred "R" :::"is_embedded_in"::: "S" means :: QUOFIELD:def 22 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" "R" "," "S" "st" (Bool (Set (Var "f")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) )); end; :: deftheorem defines :::"is_embedded_in"::: QUOFIELD:def 22 : (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "R")) ($#r1_quofield :::"is_embedded_in"::: ) (Set (Var "S"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Set (Var "f")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) )) ")" )); definitionlet "R", "S" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; pred "R" :::"is_ringisomorph_to"::: "S" means :: QUOFIELD:def 23 (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" "R" "," "S" "st" (Bool (Set (Var "f")) "is" ($#v4_quofield :::"RingIsomorphism"::: ) )); symmetry (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "st" (Bool (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Set (Var "f")) "is" ($#v4_quofield :::"RingIsomorphism"::: ) ))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "S")) "," (Set (Var "R")) "st" (Bool (Set (Var "f")) "is" ($#v4_quofield :::"RingIsomorphism"::: ) ))) ; end; :: deftheorem defines :::"is_ringisomorph_to"::: QUOFIELD:def 23 : (Bool "for" (Set (Var "R")) "," (Set (Var "S")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "R")) ($#r2_quofield :::"is_ringisomorph_to"::: ) (Set (Var "S"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "R")) "," (Set (Var "S")) "st" (Bool (Set (Var "f")) "is" ($#v4_quofield :::"RingIsomorphism"::: ) )) ")" )); begin definitionlet "I" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) ; let "x", "y" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "I")); assume (Bool (Set (Const "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Const "I")))) ; func :::"quotient"::: "(" "x" "," "y" ")" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_quofield :::"Q."::: ) "I") equals :: QUOFIELD:def 24 (Set ($#k1_domain_1 :::"["::: ) "x" "," "y" ($#k1_domain_1 :::"]"::: ) ); end; :: deftheorem defines :::"quotient"::: QUOFIELD:def 24 : (Bool "for" (Set (Var "I")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "I")) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "I"))))) "holds" (Bool (Set ($#k21_quofield :::"quotient"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_domain_1 :::"["::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k1_domain_1 :::"]"::: ) )))); definitionlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); func :::"canHom"::: "I" -> ($#m1_subset_1 :::"Function":::) "of" "I" "," (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) "I" ")" ) means :: QUOFIELD:def 25 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "I" "holds" (Bool (Set it ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k10_quofield :::"QClass."::: ) (Set "(" ($#k21_quofield :::"quotient"::: ) "(" (Set (Var "x")) "," (Set "(" ($#k5_struct_0 :::"1."::: ) "I" ")" ) ")" ")" )))); end; :: deftheorem defines :::"canHom"::: QUOFIELD:def 25 : (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "I")) "," (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k22_quofield :::"canHom"::: ) (Set (Var "I")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "I")) "holds" (Bool (Set (Set (Var "b2")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k10_quofield :::"QClass."::: ) (Set "(" ($#k21_quofield :::"quotient"::: ) "(" (Set (Var "x")) "," (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "I")) ")" ) ")" ")" )))) ")" ))); theorem :: QUOFIELD:56 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set ($#k22_quofield :::"canHom"::: ) (Set (Var "I"))) "is" ($#v1_quofield :::"RingHomomorphism"::: ) )) ; theorem :: QUOFIELD:57 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set ($#k22_quofield :::"canHom"::: ) (Set (Var "I"))) "is" ($#v3_quofield :::"embedding"::: ) )) ; theorem :: QUOFIELD:58 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set (Var "I")) ($#r1_quofield :::"is_embedded_in"::: ) (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I"))))) ; theorem :: QUOFIELD:59 (Bool "for" (Set (Var "F")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set (Var "F")) ($#r2_quofield :::"is_ringisomorph_to"::: ) (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "F"))))) ; registrationlet "I" be ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::); cluster (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) "I") -> ($#v36_algstr_0 :::"strict"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ; end; theorem :: QUOFIELD:60 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set "(" ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I")) ")" )) ($#r2_quofield :::"is_ringisomorph_to"::: ) (Set ($#k19_quofield :::"the_Field_of_Quotients"::: ) (Set (Var "I"))))) ; definitionlet "I", "F" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "I")) "," (Set (Const "F")); pred "I" :::"has_Field_of_Quotients_Pair"::: "F" "," "f" means :: QUOFIELD:def 26 (Bool "(" (Bool "f" "is" ($#v3_quofield :::"RingMonomorphism"::: ) ) & (Bool "(" "for" (Set (Var "F9")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#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"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f9")) "being" ($#m1_subset_1 :::"Function":::) "of" "I" "," (Set (Var "F9")) "st" (Bool (Bool (Set (Var "f9")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) )) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" "F" "," (Set (Var "F9")) "st" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool (Set (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) "f") ($#r2_funct_2 :::"="::: ) (Set (Var "f9"))) & (Bool "(" "for" (Set (Var "h9")) "being" ($#m1_subset_1 :::"Function":::) "of" "F" "," (Set (Var "F9")) "st" (Bool (Bool (Set (Var "h9")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool (Set (Set (Var "h9")) ($#k1_partfun1 :::"*"::: ) "f") ($#r2_funct_2 :::"="::: ) (Set (Var "f9")))) "holds" (Bool (Set (Var "h9")) ($#r2_funct_2 :::"="::: ) (Set (Var "h"))) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"has_Field_of_Quotients_Pair"::: QUOFIELD:def 26 : (Bool "for" (Set (Var "I")) "," (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "I")) "," (Set (Var "F")) "holds" (Bool "(" (Bool (Set (Var "I")) ($#r3_quofield :::"has_Field_of_Quotients_Pair"::: ) (Set (Var "F")) "," (Set (Var "f"))) "iff" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) ) & (Bool "(" "for" (Set (Var "F9")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f9")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "I")) "," (Set (Var "F9")) "st" (Bool (Bool (Set (Var "f9")) "is" ($#v3_quofield :::"RingMonomorphism"::: ) )) "holds" (Bool "ex" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "F")) "," (Set (Var "F9")) "st" (Bool "(" (Bool (Set (Var "h")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool (Set (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) ($#r2_funct_2 :::"="::: ) (Set (Var "f9"))) & (Bool "(" "for" (Set (Var "h9")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "F")) "," (Set (Var "F9")) "st" (Bool (Bool (Set (Var "h9")) "is" ($#v1_quofield :::"RingHomomorphism"::: ) ) & (Bool (Set (Set (Var "h9")) ($#k1_partfun1 :::"*"::: ) (Set (Var "f"))) ($#r2_funct_2 :::"="::: ) (Set (Var "f9")))) "holds" (Bool (Set (Var "h9")) ($#r2_funct_2 :::"="::: ) (Set (Var "h"))) ")" ) ")" ))) ")" ) ")" ) ")" ))); theorem :: QUOFIELD:61 (Bool "for" (Set (Var "I")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "ex" (Set (Var "F")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "ex" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "I")) "," (Set (Var "F")) "st" (Bool (Set (Var "I")) ($#r3_quofield :::"has_Field_of_Quotients_Pair"::: ) (Set (Var "F")) "," (Set (Var "f")))))) ; theorem :: QUOFIELD:62 (Bool "for" (Set (Var "I")) "being" ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_2 :::"domRing-like"::: ) ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "F")) "," (Set (Var "F9")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v33_algstr_0 :::"almost_left_invertible"::: ) ($#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"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "I")) "," (Set (Var "F")) (Bool "for" (Set (Var "f9")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "I")) "," (Set (Var "F9")) "st" (Bool (Bool (Set (Var "I")) ($#r3_quofield :::"has_Field_of_Quotients_Pair"::: ) (Set (Var "F")) "," (Set (Var "f"))) & (Bool (Set (Var "I")) ($#r3_quofield :::"has_Field_of_Quotients_Pair"::: ) (Set (Var "F9")) "," (Set (Var "f9")))) "holds" (Bool (Set (Var "F")) ($#r2_quofield :::"is_ringisomorph_to"::: ) (Set (Var "F9"))))))) ;