:: POLYNOM3 semantic presentation begin theorem :: POLYNOM3:1 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "st" (Bool (Bool "(" "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "L")))) ")" )) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "L")))))) ; theorem :: POLYNOM3:2 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "holds" (Bool (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "p")) ")" ))))) ; theorem :: POLYNOM3:3 (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" ($#k4_finseq_5 :::"Rev"::: ) (Set (Var "p")) ")" )))) ; theorem :: POLYNOM3:4 (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::">="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set (Var "i")))))) ; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "i" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Const "D")); :: original: :::"Del"::: redefine func :::"Del"::: "(" "p" "," "i" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" "D"; end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "a", "b" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "D")); :: original: :::"<*"::: redefine func :::"<*":::"a" "," "b":::"*>"::: -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Num 2) ($#k4_finseq_2 :::"-tuples_on"::: ) "D"); end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "k", "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "k")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Const "D"))); let "q" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Const "D"))); :: original: :::"^"::: redefine func "p" :::"^"::: "q" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "(" "k" ($#k2_nat_1 :::"+"::: ) "n" ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) "D"); end; definitionlet "D" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "k", "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Const "k")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Const "D"))); let "q" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set (Const "D"))); :: original: :::"^^"::: redefine func "p" :::"^^"::: "q" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "(" (Set "(" "k" ($#k2_nat_1 :::"+"::: ) "n" ")" ) ($#k4_finseq_2 :::"-tuples_on"::: ) "D" ")" ) ($#k3_finseq_2 :::"*"::: ) ); end; scheme :: POLYNOM3:sch 1 SeqOfSeqLambdaD{ F1() -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) , F2() -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F3( ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F4( ($#m1_hidden :::"set"::: ) "," ($#m1_hidden :::"set"::: ) ) -> ($#m1_subset_1 :::"Element"::: ) "of" (Set F1 "(" ")" ) } : (Bool "ex" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set F1 "(" ")" ) ($#k3_finseq_2 :::"*"::: ) ) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set F2 "(" ")" )) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set F2 "(" ")" )))) "holds" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" (Set (Var "p")) ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" )) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "k")) ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "p")) ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set F4 "(" (Set (Var "k")) "," (Set (Var "n")) ")" )) ")" ) ")" ) ")" ) ")" )) proof end; begin definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p", "q" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); pred "p" :::"<"::: "q" means :: POLYNOM3:def 1 (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) "n")) & (Bool (Set "p" ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<"::: ) (Set "q" ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "holds" (Bool (Set "p" ($#k1_recdef_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set "q" ($#k1_recdef_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" )); asymmetry (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Const "n")))) & (Bool (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "holds" (Bool (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" ))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "not" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Const "n")))) "or" "not" (Bool (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) "or" (Bool "ex" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i"))) & (Bool (Bool "not" (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k"))))) ")" )) ")" ))) ; end; :: deftheorem defines :::"<"::: POLYNOM3:def 1 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_polynom3 :::"<"::: ) (Set (Var "q"))) "iff" (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "holds" (Bool (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" )) ")" ))); notationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p", "q" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); synonym "q" :::">"::: "p" for "p" :::"<"::: "q"; end; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p", "q" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); pred "p" :::"<="::: "q" means :: POLYNOM3:def 2 (Bool "(" (Bool "p" ($#r1_polynom3 :::"<"::: ) "q") "or" (Bool "p" ($#r1_hidden :::"="::: ) "q") ")" ); reflexivity (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r1_polynom3 :::"<"::: ) (Set (Var "p"))) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) ")" )) ; end; :: deftheorem defines :::"<="::: POLYNOM3:def 2 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_polynom3 :::"<="::: ) (Set (Var "q"))) "iff" (Bool "(" (Bool (Set (Var "p")) ($#r1_polynom3 :::"<"::: ) (Set (Var "q"))) "or" (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q"))) ")" ) ")" ))); notationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "p", "q" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Const "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); synonym "q" :::">="::: "p" for "p" :::"<="::: "q"; end; theorem :: POLYNOM3:5 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "p")) ($#r1_polynom3 :::"<"::: ) (Set (Var "q"))) & (Bool (Set (Var "q")) ($#r1_polynom3 :::"<"::: ) (Set (Var "r")))) "implies" (Bool (Set (Var "p")) ($#r1_polynom3 :::"<"::: ) (Set (Var "r"))) ")" & "(" (Bool (Bool "(" (Bool "(" (Bool (Set (Var "p")) ($#r1_polynom3 :::"<"::: ) (Set (Var "q"))) & (Bool (Set (Var "q")) ($#r2_polynom3 :::"<="::: ) (Set (Var "r"))) ")" ) "or" (Bool "(" (Bool (Set (Var "p")) ($#r2_polynom3 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "q")) ($#r1_polynom3 :::"<"::: ) (Set (Var "r"))) ")" ) "or" (Bool "(" (Bool (Set (Var "p")) ($#r2_polynom3 :::"<="::: ) (Set (Var "q"))) & (Bool (Set (Var "q")) ($#r2_polynom3 :::"<="::: ) (Set (Var "r"))) ")" ) ")" )) "implies" (Bool (Set (Var "p")) ($#r2_polynom3 :::"<="::: ) (Set (Var "r"))) ")" ")" ))) ; theorem :: POLYNOM3:6 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"<>"::: ) (Set (Var "q")))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool "(" (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "n")))) & (Bool (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "i")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k"))) & (Bool (Set (Var "k")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "i")))) "holds" (Bool (Set (Set (Var "p")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k1_recdef_1 :::"."::: ) (Set (Var "k")))) ")" ) ")" )))) ; theorem :: POLYNOM3:7 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_polynom3 :::"<="::: ) (Set (Var "q"))) "or" (Bool (Set (Var "p")) ($#r1_polynom3 :::">"::: ) (Set (Var "q"))) ")" ))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"TuplesOrder"::: "n" -> ($#m1_subset_1 :::"Order":::) "of" (Set "(" "n" ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) means :: POLYNOM3:def 3 (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set "n" ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "p")) ($#r2_polynom3 :::"<="::: ) (Set (Var "q"))) ")" )); end; :: deftheorem defines :::"TuplesOrder"::: POLYNOM3:def 3 : (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#m1_subset_1 :::"Order":::) "of" (Set "(" (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_polynom3 :::"TuplesOrder"::: ) (Set (Var "n")))) "iff" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "n")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set ($#k4_tarski :::"["::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k4_tarski :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "p")) ($#r2_polynom3 :::"<="::: ) (Set (Var "q"))) ")" )) ")" ))); registrationlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k5_polynom3 :::"TuplesOrder"::: ) "n") -> ($#v3_orders_1 :::"being_linear-order"::: ) ; end; begin definitionlet "i" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"Decomp"::: "(" "n" "," "i" ")" -> ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "i" ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) means :: POLYNOM3:def 4 (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" "i" ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set ($#k7_pre_poly :::"SgmX"::: ) "(" (Set "(" ($#k5_polynom3 :::"TuplesOrder"::: ) "i" ")" ) "," (Set (Var "A")) ")" )) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set "i" ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "iff" (Bool (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) "n") ")" ) ")" ) ")" )); end; :: deftheorem defines :::"Decomp"::: POLYNOM3:def 4 : (Bool "for" (Set (Var "i")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b3")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Set (Var "i")) ")" )) "iff" (Bool "ex" (Set (Var "A")) "being" ($#v1_finset_1 :::"finite"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) ")" ) "st" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k7_pre_poly :::"SgmX"::: ) "(" (Set "(" ($#k5_polynom3 :::"TuplesOrder"::: ) (Set (Var "i")) ")" ) "," (Set (Var "A")) ")" )) & (Bool "(" "for" (Set (Var "p")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set (Var "i")) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "holds" (Bool "(" (Bool (Set (Var "p")) ($#r2_hidden :::"in"::: ) (Set (Var "A"))) "iff" (Bool (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set (Var "p"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) ")" ) ")" ) ")" )) ")" )))); registrationlet "i" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k6_polynom3 :::"Decomp"::: ) "(" "n" "," "i" ")" ) -> ($#v2_funct_1 :::"one-to-one"::: ) ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_pre_poly :::"FinSequence-yielding"::: ) ; end; theorem :: POLYNOM3:8 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 1) ")" ")" )) ($#r1_hidden :::"="::: ) (Num 1))) ; theorem :: POLYNOM3:9 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 2) ")" ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) ; theorem :: POLYNOM3:10 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_matrlin :::"<*"::: ) (Set ($#k3_pre_poly :::"<*"::: ) (Set (Var "n")) ($#k3_pre_poly :::"*>"::: ) ) ($#k2_matrlin :::"*>"::: ) ))) ; theorem :: POLYNOM3:11 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "n")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_polynom3 :::"<*"::: ) (Set (Var "k1")) "," (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k1")) ")" ) ($#k2_polynom3 :::"*>"::: ) )) & (Bool (Set (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set ($#k2_polynom3 :::"<*"::: ) (Set (Var "k2")) "," (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k2")) ")" ) ($#k2_polynom3 :::"*>"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "j"))) "iff" (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "k2"))) ")" )) ; theorem :: POLYNOM3:12 (Bool "for" (Set (Var "i")) "," (Set (Var "n")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_polynom3 :::"<*"::: ) (Set (Var "k1")) "," (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k1")) ")" ) ($#k2_polynom3 :::"*>"::: ) )) & (Bool (Set (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_polynom3 :::"<*"::: ) (Set (Var "k2")) "," (Set "(" (Set (Var "n")) ($#k7_nat_d :::"-'"::: ) (Set (Var "k2")) ")" ) ($#k2_polynom3 :::"*>"::: ) ))) "holds" (Bool (Set (Var "k2")) ($#r1_hidden :::"="::: ) (Set (Set (Var "k1")) ($#k2_nat_1 :::"+"::: ) (Num 1)))) ; theorem :: POLYNOM3:13 (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set ($#k2_polynom3 :::"<*"::: ) (Set ($#k6_numbers :::"0"::: ) ) "," (Set (Var "n")) ($#k2_polynom3 :::"*>"::: ) ))) ; theorem :: POLYNOM3:14 (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Set "(" ($#k6_polynom3 :::"Decomp"::: ) "(" (Set (Var "n")) "," (Num 2) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_polynom3 :::"<*"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) "," (Set "(" (Set "(" (Set (Var "n")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ($#k2_polynom3 :::"*>"::: ) ))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) ; let "p", "q", "r" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "L")); let "t" be ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Num 3) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); func :::"prodTuples"::: "(" "p" "," "q" "," "r" "," "t" ")" -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k3_finseq_2 :::"*"::: ) ) means :: POLYNOM3:def 5 (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) "t")) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) "t"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" "p" ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" "t" ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "q" ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" "t" ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 2) ")" ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "r" ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" "t" ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 3) ")" ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"prodTuples"::: POLYNOM3:def 5 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Num 3) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "b6")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) ($#k3_finseq_2 :::"*"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set ($#k7_polynom3 :::"prodTuples"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "t")) ")" )) "iff" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "b6"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "t")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "t"))))) "holds" (Bool (Set (Set (Var "b6")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "p")) ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "t")) ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "q")) ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "t")) ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 2) ")" ) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "r")) ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "t")) ($#k9_pre_poly :::"/."::: ) (Set (Var "k")) ")" ) ($#k7_partfun1 :::"/."::: ) (Num 3) ")" ) ")" ))) ")" ) ")" ) ")" ))))); theorem :: POLYNOM3:15 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l3_algstr_0 :::"multMagma"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "t")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Num 3) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Permutation":::) "of" (Set "(" ($#k4_finseq_1 :::"dom"::: ) (Set (Var "t")) ")" ) (Bool "for" (Set (Var "t1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Num 3) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )) "st" (Bool (Bool (Set (Var "t1")) ($#r1_hidden :::"="::: ) (Set (Set (Var "t")) ($#k1_partfun1 :::"*"::: ) (Set (Var "P"))))) "holds" (Bool (Set ($#k7_polynom3 :::"prodTuples"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "t1")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_polynom3 :::"prodTuples"::: ) "(" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "," (Set (Var "t")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set (Var "P"))))))))) ; theorem :: POLYNOM3:16 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D")) ($#k3_finseq_2 :::"*"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k10_pre_poly :::"Card"::: ) (Set "(" (Set (Var "f")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_pre_poly :::"Card"::: ) (Set (Var "f")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))))))) ; theorem :: POLYNOM3:17 (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "q")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) "holds" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))))))) ; theorem :: POLYNOM3:18 (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "j")) ")" ))))) ; theorem :: POLYNOM3:19 (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Var "D")) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) ($#k7_finseq_1 :::"^"::: ) (Set ($#k9_finseq_1 :::"<*"::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ($#k9_finseq_1 :::"*>"::: ) )))))) ; theorem :: POLYNOM3:20 (Bool "for" (Set (Var "p")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p"))))) "holds" (Bool (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) ")" ) ($#k2_xcmplx_0 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ))))) ; theorem :: POLYNOM3:21 (Bool "for" (Set (Var "p")) "being" ($#m1_trees_4 :::"FinSequence"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "," (Set (Var "k1")) "," (Set (Var "k2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "p")))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k1"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "k2"))) & (Bool (Set (Var "k1")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Var "k2")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "p")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "j")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ))) & (Bool (Set (Set "(" ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i")) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k1"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" (Set (Var "p")) ($#k17_finseq_1 :::"|"::: ) (Set (Var "j")) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "k2"))))) "holds" (Bool "(" (Bool (Set (Var "i")) ($#r1_hidden :::"="::: ) (Set (Var "j"))) & (Bool (Set (Var "k1")) ($#r1_hidden :::"="::: ) (Set (Var "k2"))) ")" ))) ; theorem :: POLYNOM3:22 (Bool "for" (Set (Var "D1")) "," (Set (Var "D2")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D1")) ($#k3_finseq_2 :::"*"::: ) ) (Bool "for" (Set (Var "f2")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set (Set (Var "D2")) ($#k3_finseq_2 :::"*"::: ) ) (Bool "for" (Set (Var "i1")) "," (Set (Var "i2")) "," (Set (Var "j1")) "," (Set (Var "j2")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i1")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f1")))) & (Bool (Set (Var "i2")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f2")))) & (Bool (Set (Var "j1")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k1_funct_1 :::"."::: ) (Set (Var "i1")) ")" ))) & (Bool (Set (Var "j2")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k1_funct_1 :::"."::: ) (Set (Var "i2")) ")" ))) & (Bool (Set ($#k10_pre_poly :::"Card"::: ) (Set (Var "f1"))) ($#r1_hidden :::"="::: ) (Set ($#k10_pre_poly :::"Card"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" (Set "(" ($#k10_pre_poly :::"Card"::: ) (Set (Var "f1")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i1")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "j1"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_wsierp_1 :::"Sum"::: ) (Set "(" (Set "(" ($#k10_pre_poly :::"Card"::: ) (Set (Var "f2")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i2")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ")" ) ($#k2_nat_1 :::"+"::: ) (Set (Var "j2"))))) "holds" (Bool "(" (Bool (Set (Var "i1")) ($#r1_hidden :::"="::: ) (Set (Var "i2"))) & (Bool (Set (Var "j1")) ($#r1_hidden :::"="::: ) (Set (Var "j2"))) ")" ))))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) ; mode Polynomial of "L" is ($#m1_subset_1 :::"AlgSequence":::) "of" "L"; end; theorem :: POLYNOM3:23 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" (Bool (Set (Var "n")) ($#r1_xxreal_0 :::">="::: ) (Set ($#k1_algseq_1 :::"len"::: ) (Set (Var "p")))) "iff" (Bool (Set (Var "n")) ($#r1_algseq_1 :::"is_at_least_length_of"::: ) (Set (Var "p"))) ")" )))) ; scheme :: POLYNOM3:sch 2 PolynomialLambdaF{ F1() -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) , F2() -> ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ), F3( ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) )) -> ($#m1_subset_1 :::"Element":::) "of" (Set F1 "(" ")" ) } : (Bool "ex" (Set (Var "p")) "being" ($#m1_subset_1 :::"Polynomial":::) "of" (Set F1 "(" ")" ) "st" (Bool "(" (Bool (Set ($#k1_algseq_1 :::"len"::: ) (Set (Var "p"))) ($#r1_xxreal_0 :::"<="::: ) (Set F2 "(" ")" )) & (Bool "(" "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_xxreal_0 :::"<"::: ) (Set F2 "(" ")" ))) "holds" (Bool (Set (Set (Var "p")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set F3 "(" (Set (Var "n")) ")" )) ")" ) ")" )) proof end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "p", "q" be ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Const "L")); cluster (Set "p" ($#k2_normsp_1 :::"+"::: ) "q") -> ($#v1_algseq_1 :::"finite-Support"::: ) ; end; theorem :: POLYNOM3:24 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "n")) ($#r1_algseq_1 :::"is_at_least_length_of"::: ) (Set (Var "p"))) & (Bool (Set (Var "n")) ($#r1_algseq_1 :::"is_at_least_length_of"::: ) (Set (Var "q")))) "holds" (Bool (Set (Var "n")) ($#r1_algseq_1 :::"is_at_least_length_of"::: ) (Set (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "q"))))))) ; theorem :: POLYNOM3:25 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Var "L")) "holds" (Bool (Set ($#k2_algseq_1 :::"support"::: ) (Set "(" (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "q")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k2_algseq_1 :::"support"::: ) (Set (Var "p")) ")" ) ($#k4_subset_1 :::"\/"::: ) (Set "(" ($#k2_algseq_1 :::"support"::: ) (Set (Var "q")) ")" ))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "p", "q" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "L")); :: original: :::"+"::: redefine func "p" :::"+"::: "q" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k5_numbers :::"NAT"::: ) ) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") ($#k2_zfmisc_1 :::":]"::: ) )); commutativity (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "L")) "holds" (Bool (Set (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "p"))))) ; end; theorem :: POLYNOM3:26 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "q")) ")" ) ($#k2_normsp_1 :::"+"::: ) (Set (Var "r"))) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "r")) ")" ))))) ; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "p" be ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Const "L")); cluster (Set ($#k5_vfunct_1 :::"-"::: ) "p") -> ($#v1_algseq_1 :::"finite-Support"::: ) ; end; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "p", "q" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "L")); redefine func "p" :::"-"::: "q" equals :: POLYNOM3:def 6 (Set "p" ($#k2_normsp_1 :::"+"::: ) (Set "(" ($#k5_vfunct_1 :::"-"::: ) "q" ")" )); end; :: deftheorem defines :::"-"::: POLYNOM3:def 6 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "p")) ($#k3_normsp_1 :::"-"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "q")) ")" ))))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) ; let "p", "q" be ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Const "L")); cluster (Set "p" ($#k3_normsp_1 :::"-"::: ) "q") -> ($#v1_algseq_1 :::"finite-Support"::: ) ; end; theorem :: POLYNOM3:27 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) (Bool "for" (Set (Var "n")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k3_normsp_1 :::"-"::: ) (Set (Var "q")) ")" ) ($#k1_normsp_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "q")) ($#k1_normsp_1 :::"."::: ) (Set (Var "n")) ")" )))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) ; func :::"0_."::: "L" -> ($#m1_subset_1 :::"sequence":::) "of" "L" equals :: POLYNOM3:def 7 (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "L" ")" )); end; :: deftheorem defines :::"0_."::: POLYNOM3:def 7 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) "holds" (Bool (Set ($#k9_polynom3 :::"0_."::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set ($#k5_numbers :::"NAT"::: ) ) ($#k8_funcop_1 :::"-->"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "L")) ")" )))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l2_struct_0 :::"ZeroStr"::: ) ; cluster (Set ($#k9_polynom3 :::"0_."::: ) "L") -> ($#v1_algseq_1 :::"finite-Support"::: ) ; end; theorem :: POLYNOM3:28 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set "(" ($#k9_polynom3 :::"0_."::: ) (Set (Var "L")) ")" )) ($#r2_funct_2 :::"="::: ) (Set (Var "p"))))) ; theorem :: POLYNOM3:29 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l2_algstr_0 :::"addLoopStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "p")) ($#k3_normsp_1 :::"-"::: ) (Set (Var "p"))) ($#r2_funct_2 :::"="::: ) (Set ($#k9_polynom3 :::"0_."::: ) (Set (Var "L")))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; func :::"1_."::: "L" -> ($#m1_subset_1 :::"sequence":::) "of" "L" equals :: POLYNOM3:def 8 (Set (Set "(" ($#k9_polynom3 :::"0_."::: ) "L" ")" ) ($#k15_funct_7 :::"+*"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" ($#k5_struct_0 :::"1."::: ) "L" ")" ) ")" ); end; :: deftheorem defines :::"1_."::: POLYNOM3:def 8 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) "holds" (Bool (Set ($#k10_polynom3 :::"1_."::: ) (Set (Var "L"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_polynom3 :::"0_."::: ) (Set (Var "L")) ")" ) ($#k15_funct_7 :::"+*"::: ) "(" (Set ($#k6_numbers :::"0"::: ) ) "," (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "L")) ")" ) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) ; cluster (Set ($#k10_polynom3 :::"1_."::: ) "L") -> ($#v1_algseq_1 :::"finite-Support"::: ) ; end; theorem :: POLYNOM3:30 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l5_algstr_0 :::"multLoopStr_0"::: ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k10_polynom3 :::"1_."::: ) (Set (Var "L")) ")" ) ($#k1_normsp_1 :::"."::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "L")))) & (Bool "(" "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k10_polynom3 :::"1_."::: ) (Set (Var "L")) ")" ) ($#k8_nat_1 :::"."::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "L")))) ")" ) ")" )) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "p", "q" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "L")); func "p" :::"*'"::: "q" -> ($#m1_subset_1 :::"sequence":::) "of" "L" means :: POLYNOM3:def 9 (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "r")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "L") "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set it ($#k1_normsp_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "r"))))) "holds" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "p" ($#k1_normsp_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" "q" ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "k")) ")" ) ")" ))) ")" ) ")" ))); end; :: deftheorem defines :::"*'"::: POLYNOM3:def 9 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "b4")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "q")))) "iff" (Bool "for" (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "ex" (Set (Var "r")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "L"))) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "r"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1))) & (Bool (Set (Set (Var "b4")) ($#k1_normsp_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_rlvect_1 :::"Sum"::: ) (Set (Var "r")))) & (Bool "(" "for" (Set (Var "k")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "k")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "r"))))) "holds" (Bool (Set (Set (Var "r")) ($#k1_funct_1 :::"."::: ) (Set (Var "k"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k1_normsp_1 :::"."::: ) (Set "(" (Set (Var "k")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" (Set (Var "q")) ($#k1_normsp_1 :::"."::: ) (Set "(" (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "k")) ")" ) ")" ))) ")" ) ")" ))) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "p", "q" be ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Const "L")); cluster (Set "p" ($#k11_polynom3 :::"*'"::: ) "q") -> ($#v1_algseq_1 :::"finite-Support"::: ) ; end; theorem :: POLYNOM3:31 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set "(" (Set (Var "q")) ($#k8_polynom3 :::"+"::: ) (Set (Var "r")) ")" )) ($#r2_funct_2 :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "q")) ")" ) ($#k8_polynom3 :::"+"::: ) (Set "(" (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "r")) ")" ))))) ; theorem :: POLYNOM3:32 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_vectsp_1 :::"left-distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k8_polynom3 :::"+"::: ) (Set (Var "q")) ")" ) ($#k11_polynom3 :::"*'"::: ) (Set (Var "r"))) ($#r2_funct_2 :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "r")) ")" ) ($#k8_polynom3 :::"+"::: ) (Set "(" (Set (Var "q")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "r")) ")" ))))) ; definitionlet "n" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); :: original: :::"<*"::: redefine func :::"<*":::"n":::"*>"::: -> ($#m2_finseq_2 :::"Element"::: ) "of" (Set (Num 1) ($#k4_finseq_2 :::"-tuples_on"::: ) (Set ($#k5_numbers :::"NAT"::: ) )); end; theorem :: POLYNOM3:33 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set "(" (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "q")) ")" ) ($#k11_polynom3 :::"*'"::: ) (Set (Var "r"))) ($#r2_funct_2 :::"="::: ) (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set "(" (Set (Var "q")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "r")) ")" ))))) ; definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_group_1 :::"commutative"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "p", "q" be ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "L")); :: original: :::"*'"::: redefine func "p" :::"*'"::: "q" -> ($#m1_subset_1 :::"sequence":::) "of" "L"; commutativity (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Const "L")) "holds" (Bool (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "q"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "q")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "p"))))) ; end; theorem :: POLYNOM3:34 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set "(" ($#k9_polynom3 :::"0_."::: ) (Set (Var "L")) ")" )) ($#r2_funct_2 :::"="::: ) (Set ($#k9_polynom3 :::"0_."::: ) (Set (Var "L")))))) ; theorem :: POLYNOM3:35 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "holds" (Bool (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set "(" ($#k10_polynom3 :::"1_."::: ) (Set (Var "L")) ")" )) ($#r2_funct_2 :::"="::: ) (Set (Var "p"))))) ; begin definitionlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; func :::"Polynom-Ring"::: "L" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) means :: POLYNOM3:def 10 (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it)) "iff" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Polynomial":::) "of" "L") ")" ) ")" ) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" it (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" "L" "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "q"))))) ")" ) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" it (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" "L" "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "q"))))) ")" ) & (Bool (Set ($#k4_struct_0 :::"0."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k9_polynom3 :::"0_."::: ) "L")) & (Bool (Set ($#k5_struct_0 :::"1."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k10_polynom3 :::"1_."::: ) "L")) ")" ); end; :: deftheorem defines :::"Polynom-Ring"::: POLYNOM3:def 10 : (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "b2")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) (Set (Var "L")))) "iff" (Bool "(" (Bool "(" "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2")))) "iff" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Polynomial":::) "of" (Set (Var "L"))) ")" ) ")" ) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2")) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set (Var "x")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k2_normsp_1 :::"+"::: ) (Set (Var "q"))))) ")" ) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "b2")) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "L")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "p"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "q")))) "holds" (Bool (Set (Set (Var "x")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "p")) ($#k11_polynom3 :::"*'"::: ) (Set (Var "q"))))) ")" ) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k9_polynom3 :::"0_."::: ) (Set (Var "L")))) & (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k10_polynom3 :::"1_."::: ) (Set (Var "L")))) ")" ) ")" ))); registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v36_algstr_0 :::"strict"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v5_group_1 :::"commutative"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v3_group_1 :::"associative"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ; end; registrationlet "L" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; cluster (Set ($#k14_polynom3 :::"Polynom-Ring"::: ) "L") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ; end; theorem :: POLYNOM3:36 (Bool "for" (Set (Var "L")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v4_vectsp_1 :::"well-unital"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set "(" ($#k14_polynom3 :::"Polynom-Ring"::: ) (Set (Var "L")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k10_polynom3 :::"1_."::: ) (Set (Var "L"))))) ;