:: FVALUAT1 semantic presentation begin theorem :: FVALUAT1:1 (Bool "for" (Set (Var "x")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k2_xxreal_3 :::"-"::: ) (Set (Var "x"))))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: FVALUAT1:2 (Bool "for" (Set (Var "x")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Set (Var "x")) ($#k1_xxreal_3 :::"+"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: FVALUAT1:3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "s")) "," (Set (Var "z")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "x"))) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "y"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "s"))) & (Bool (Set (Var "s")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "z")))) "holds" (Bool (Set (Set (Var "x")) ($#k4_xxreal_3 :::"*"::: ) (Set (Var "s"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "y")) ($#k4_xxreal_3 :::"*"::: ) (Set (Var "z"))))) ; theorem :: FVALUAT1:4 (Bool "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "x")) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "y"))))) ; theorem :: FVALUAT1:5 (Bool "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set (Var "x")) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: FVALUAT1:6 (Bool "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) )) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x"))) & (Bool (Set (Var "y")) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "x")) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "y"))) ($#r1_xxreal_0 :::"<"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; theorem :: FVALUAT1:7 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool "(" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ")" ) "or" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set ($#k1_numbers :::"REAL"::: ) )) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k1_xxreal_3 :::"+"::: ) (Set (Var "y")) ")" ) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "x")) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "z")) ")" ) ($#k1_xxreal_3 :::"+"::: ) (Set "(" (Set (Var "y")) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "z")) ")" )))) ; theorem :: FVALUAT1:8 (Bool "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "x")) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "y")) ")" ) ($#k4_xxreal_3 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "x")))) ; theorem :: FVALUAT1:9 (Bool "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#v1_xxreal_0 :::"ext-real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k2_xxreal_0 :::"-infty"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "x")) ($#k6_xxreal_3 :::"/"::: ) (Set (Var "y"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) ; definitionlet "x" be ($#m1_hidden :::"number"::: ) ; attr "x" is :::"ext-integer"::: means :: FVALUAT1:def 1 (Bool "(" (Bool "x" "is" ($#v1_int_1 :::"integer"::: ) ) "or" (Bool "x" ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) ")" ); end; :: deftheorem defines :::"ext-integer"::: FVALUAT1:def 1 : (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"number"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_fvaluat1 :::"ext-integer"::: ) ) "iff" (Bool "(" (Bool (Set (Var "x")) "is" ($#v1_int_1 :::"integer"::: ) ) "or" (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) ")" ) ")" )); registration cluster ($#v1_fvaluat1 :::"ext-integer"::: ) -> ($#v1_xxreal_0 :::"ext-real"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster (Set ($#k1_xxreal_0 :::"+infty"::: ) ) -> ($#v1_fvaluat1 :::"ext-integer"::: ) ; cluster (Set ($#k2_xxreal_0 :::"-infty"::: ) ) -> ($#~v1_fvaluat1 "non" ($#v1_fvaluat1 :::"ext-integer"::: ) ) ; cluster (Set ($#k8_mesfunc1 :::"1."::: ) ) -> ($#v1_xreal_0 :::"real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#v1_fvaluat1 :::"ext-integer"::: ) ; cluster ($#v1_int_1 :::"integer"::: ) -> ($#v1_fvaluat1 :::"ext-integer"::: ) for ($#m1_hidden :::"set"::: ) ; cluster ($#v1_xreal_0 :::"real"::: ) ($#v1_fvaluat1 :::"ext-integer"::: ) -> ($#v1_int_1 :::"integer"::: ) for ($#m1_hidden :::"set"::: ) ; end; registration cluster ($#v1_xreal_0 :::"real"::: ) ($#v1_xxreal_0 :::"ext-real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#v1_fvaluat1 :::"ext-integer"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k7_numbers :::"ExtREAL"::: ) ); cluster ($#v1_xxreal_0 :::"ext-real"::: ) ($#v2_xxreal_0 :::"positive"::: ) ($#v1_fvaluat1 :::"ext-integer"::: ) for ($#m1_hidden :::"set"::: ) ; end; definitionmode ExtInt is ($#v1_fvaluat1 :::"ext-integer"::: ) ($#m1_hidden :::"number"::: ) ; end; theorem :: FVALUAT1:10 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_hidden :::"ExtInt":::) "st" (Bool (Bool (Set (Var "x")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "y")))) "holds" (Bool (Set (Set (Var "x")) ($#k1_xxreal_3 :::"+"::: ) (Num 1)) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "y")))) ; theorem :: FVALUAT1:11 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"ExtInt":::) "holds" (Bool (Set ($#k2_xxreal_0 :::"-infty"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "x")))) ; definitionlet "X" be ($#v2_membered :::"ext-real-membered"::: ) ($#m1_hidden :::"set"::: ) ; given "i0" being ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"ExtInt":::) such that (Bool (Set (Const "i0")) ($#r2_hidden :::"in"::: ) (Set (Const "X"))) ; func :::"least-positive"::: "X" -> ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"ExtInt":::) means :: FVALUAT1:def 2 (Bool "(" (Bool it ($#r2_hidden :::"in"::: ) "X") & (Bool "(" "for" (Set (Var "i")) "being" ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"ExtInt":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool it ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) ")" ) ")" ); end; :: deftheorem defines :::"least-positive"::: FVALUAT1:def 2 : (Bool "for" (Set (Var "X")) "being" ($#v2_membered :::"ext-real-membered"::: ) ($#m1_hidden :::"set"::: ) "st" (Bool (Bool "ex" (Set (Var "i0")) "being" ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"ExtInt":::) "st" (Bool (Set (Var "i0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))))) "holds" (Bool "for" (Set (Var "b2")) "being" ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"ExtInt":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_fvaluat1 :::"least-positive"::: ) (Set (Var "X")))) "iff" (Bool "(" (Bool (Set (Var "b2")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#v2_xxreal_0 :::"positive"::: ) ($#m1_hidden :::"ExtInt":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "b2")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_hidden :::"Relation":::); attr "f" is :::"e.i.-valued"::: means :: FVALUAT1:def 3 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) "f"))) "holds" (Bool (Set (Var "x")) "is" ($#v1_fvaluat1 :::"ext-integer"::: ) )); end; :: deftheorem defines :::"e.i.-valued"::: FVALUAT1:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_hidden :::"Relation":::) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v2_fvaluat1 :::"e.i.-valued"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "x")) "is" ($#v1_fvaluat1 :::"ext-integer"::: ) )) ")" )); registration cluster ($#v1_relat_1 :::"Relation-like"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v2_fvaluat1 :::"e.i.-valued"::: ) for ($#m1_hidden :::"set"::: ) ; end; registrationlet "A" be ($#m1_hidden :::"set"::: ) ; cluster ($#v1_relat_1 :::"Relation-like"::: ) "A" ($#v4_relat_1 :::"-defined"::: ) (Set ($#k7_numbers :::"ExtREAL"::: ) ) ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) bbbadV1_FUNCT_2("A" "," (Set ($#k7_numbers :::"ExtREAL"::: ) )) ($#v2_fvaluat1 :::"e.i.-valued"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) "A" "," (Set ($#k7_numbers :::"ExtREAL"::: ) ) ($#k2_zfmisc_1 :::":]"::: ) )); end; registrationlet "f" be ($#v2_fvaluat1 :::"e.i.-valued"::: ) ($#m1_hidden :::"Function":::); let "x" be ($#m1_hidden :::"set"::: ) ; cluster (Set "f" ($#k1_funct_1 :::"."::: ) "x") -> ($#v1_fvaluat1 :::"ext-integer"::: ) ; end; begin theorem :: FVALUAT1:12 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v6_vectsp_1 :::"left_unital"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool (Set (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" ) ")" ) ($#k6_algstr_0 :::"*"::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "K"))))) ; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "S" be ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "K")); let "n" be ($#m1_hidden :::"Nat":::); func "S" :::"|^"::: "n" -> ($#m1_subset_1 :::"Subset":::) "of" "K" means :: FVALUAT1:def 4 (Bool it ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K")) if (Bool "n" ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) otherwise (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "K")) "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) "n") & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) "S") & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "i")) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set "S" ($#k14_ideal_1 :::"*'"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ))) ")" ) ")" )); end; :: deftheorem defines :::"|^"::: FVALUAT1:def 4 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "b4")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k2_fvaluat1 :::"|^"::: ) (Set (Var "n")))) "iff" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "K")))) ")" ) ")" & "(" (Bool (Bool (Bool "not" (Set (Var "n")) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) "implies" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k2_fvaluat1 :::"|^"::: ) (Set (Var "n")))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "K")))) "st" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set (Var "n"))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "S"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Set (Var "i")) ($#k3_real_1 :::"+"::: ) (Num 1)) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "i")) ($#k3_real_1 :::"+"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "S")) ($#k14_ideal_1 :::"*'"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ))) ")" ) ")" )) ")" ) ")" ")" ))))); theorem :: FVALUAT1:13 (Bool "for" (Set (Var "D")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "A")) ($#k2_fvaluat1 :::"|^"::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "A"))))) ; theorem :: FVALUAT1:14 (Bool "for" (Set (Var "D")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "D")) "holds" (Bool (Set (Set (Var "A")) ($#k2_fvaluat1 :::"|^"::: ) (Num 2)) ($#r1_hidden :::"="::: ) (Set (Set (Var "A")) ($#k14_ideal_1 :::"*'"::: ) (Set (Var "A")))))) ; registrationlet "R" be ($#l6_algstr_0 :::"Ring":::); let "S" be ($#m1_subset_1 :::"Ideal":::) "of" (Set (Const "R")); let "n" be ($#m1_hidden :::"Nat":::); cluster (Set "S" ($#k2_fvaluat1 :::"|^"::: ) "n") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#v2_ideal_1 :::"left-ideal"::: ) ($#v3_ideal_1 :::"right-ideal"::: ) ; end; definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "g" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "G")); let "i" be ($#m1_hidden :::"Integer":::); func "g" :::"|^"::: "i" -> ($#m1_subset_1 :::"Element":::) "of" "G" equals :: FVALUAT1:def 5 (Set (Set "(" ($#k4_group_1 :::"power"::: ) "G" ")" ) ($#k2_binop_1 :::"."::: ) "(" "g" "," (Set "(" ($#k1_int_2 :::"abs"::: ) "i" ")" ) ")" ) if (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) "i") otherwise (Set (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) "G" ")" ) ($#k2_binop_1 :::"."::: ) "(" "g" "," (Set "(" ($#k1_int_2 :::"abs"::: ) "i" ")" ) ")" ")" ) ($#k11_algstr_0 :::"""::: ) ); end; :: deftheorem defines :::"|^"::: FVALUAT1:def 5 : (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Integer":::) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i")))) "implies" (Bool (Set (Set (Var "g")) ($#k3_fvaluat1 :::"|^"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "G")) ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "g")) "," (Set "(" ($#k1_int_2 :::"abs"::: ) (Set (Var "i")) ")" ) ")" )) ")" & "(" (Bool (Bool (Bool "not" (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))))) "implies" (Bool (Set (Set (Var "g")) ($#k3_fvaluat1 :::"|^"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "G")) ")" ) ($#k2_binop_1 :::"."::: ) "(" (Set (Var "g")) "," (Set "(" ($#k1_int_2 :::"abs"::: ) (Set (Var "i")) ")" ) ")" ")" ) ($#k11_algstr_0 :::"""::: ) )) ")" ")" )))); definitionlet "G" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "g" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "G")); let "n" be ($#m1_hidden :::"Nat":::); redefine func "g" :::"|^"::: "n" equals :: FVALUAT1:def 6 (Set (Set "(" ($#k4_group_1 :::"power"::: ) "G" ")" ) ($#k1_binop_1 :::"."::: ) "(" "g" "," "n" ")" ); end; :: deftheorem defines :::"|^"::: FVALUAT1:def 6 : (Bool "for" (Set (Var "G")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "G")) (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) "holds" (Bool (Set (Set (Var "g")) ($#k3_fvaluat1 :::"|^"::: ) (Set (Var "n"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "G")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "g")) "," (Set (Var "n")) ")" ))))); theorem :: FVALUAT1:15 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "a")) ($#k3_fvaluat1 :::"|^"::: ) (Set "(" (Set (Var "n")) ($#k2_xcmplx_0 :::"+"::: ) (Set (Var "m")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "a")) ($#k3_fvaluat1 :::"|^"::: ) (Set (Var "n")) ")" ) ($#k8_group_1 :::"*"::: ) (Set "(" (Set (Var "a")) ($#k3_fvaluat1 :::"|^"::: ) (Set (Var "m")) ")" )))))) ; theorem :: FVALUAT1:16 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "a")) ($#k3_fvaluat1 :::"|^"::: ) (Set (Var "i"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))))) ; begin definitionlet "K" be ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; attr "K" is :::"having_valuation"::: means :: FVALUAT1:def 7 (Bool "ex" (Set (Var "f")) "being" ($#v2_fvaluat1 :::"e.i.-valued"::: ) ($#m1_subset_1 :::"Function":::) "of" "K" "," (Set ($#k7_numbers :::"ExtREAL"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "K" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "K"))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_numbers :::"INT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_xxreal_3 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k5_struct_0 :::"1."::: ) "K" ")" ) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) ")" )) ")" )); end; :: deftheorem defines :::"having_valuation"::: FVALUAT1:def 7 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"doubleLoopStr"::: ) "holds" (Bool "(" (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) "iff" (Bool "ex" (Set (Var "f")) "being" ($#v2_fvaluat1 :::"e.i.-valued"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "K")) "," (Set ($#k7_numbers :::"ExtREAL"::: ) ) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_numbers :::"INT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_xxreal_3 :::"+"::: ) (Set "(" (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) ")" )) ")" )) ")" )); definitionlet "K" be ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; assume (Bool (Set (Const "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) ; mode :::"Valuation"::: "of" "K" -> ($#v2_fvaluat1 :::"e.i.-valued"::: ) ($#m1_subset_1 :::"Function":::) "of" "K" "," (Set ($#k7_numbers :::"ExtREAL"::: ) ) means :: FVALUAT1:def 8 (Bool "(" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) "K" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) "K"))) "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_numbers :::"INT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "holds" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" it ($#k1_funct_1 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_xxreal_3 :::"+"::: ) (Set "(" it ($#k1_funct_1 :::"."::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set it ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k5_struct_0 :::"1."::: ) "K" ")" ) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "st" (Bool "(" (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set it ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) ")" )) ")" ); end; :: deftheorem defines :::"Valuation"::: FVALUAT1:def 8 : (Bool "for" (Set (Var "K")) "being" ($#l6_algstr_0 :::"doubleLoopStr"::: ) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "b2")) "being" ($#v2_fvaluat1 :::"e.i.-valued"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "K")) "," (Set ($#k7_numbers :::"ExtREAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b2")) "is" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K"))) "iff" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r2_hidden :::"in"::: ) (Set ($#k4_numbers :::"INT"::: ) )) ")" ) & (Bool "(" "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set (Var "a")) ($#k6_algstr_0 :::"*"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "a")) ")" ) ($#k1_xxreal_3 :::"+"::: ) (Set "(" (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "b")) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set "(" (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" ) ($#k1_algstr_0 :::"+"::: ) (Set (Var "a")) ")" ))) ")" ) & (Bool "ex" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool "(" (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Set (Var "b2")) ($#k1_funct_1 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) ")" )) ")" ) ")" ))); theorem :: FVALUAT1:17 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: FVALUAT1:18 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"<>"::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) ))))) ; theorem :: FVALUAT1:19 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: FVALUAT1:20 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_algstr_0 :::"-"::: ) (Set (Var "a")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))))))) ; theorem :: FVALUAT1:21 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k11_algstr_0 :::"""::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_supinf_2 :::"-"::: ) (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" )))))) ; theorem :: FVALUAT1:22 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) ($#k4_supinf_2 :::"-"::: ) (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b")) ")" )))))) ; theorem :: FVALUAT1:23 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "b")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k2_supinf_2 :::"-"::: ) (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "b")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "a")) ")" ) ")" )))))) ; theorem :: FVALUAT1:24 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "b")) ")" )))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))))))) ; theorem :: FVALUAT1:25 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "b")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "b")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "a")) ")" )))))) ; theorem :: FVALUAT1:26 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "b")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "b")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b"))))))) ; theorem :: FVALUAT1:27 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k3_xxreal_0 :::"min"::: ) "(" (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) "," (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b")) ")" ) ")" ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "b")) ")" )))))) ; theorem :: FVALUAT1:28 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b"))))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "b")) ")" )))))) ; theorem :: FVALUAT1:29 (Bool "for" (Set (Var "i")) "being" ($#m1_hidden :::"Integer":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k3_fvaluat1 :::"|^"::: ) (Set (Var "i")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k4_xxreal_3 :::"*"::: ) (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ))))))) ; theorem :: FVALUAT1:30 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "a")) ")" )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))))))) ; theorem :: FVALUAT1:31 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k5_struct_0 :::"1."::: ) (Set (Var "K")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "a")) ")" )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))))))) ; theorem :: FVALUAT1:32 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b"))))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "b")) ($#k1_vectsp_2 :::"/"::: ) (Set (Var "a")) ")" )))))) ; theorem :: FVALUAT1:33 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k1_xxreal_0 :::"+infty"::: ) ) ($#r2_hidden :::"in"::: ) (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "v")))))) ; theorem :: FVALUAT1:34 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k1_fvaluat1 :::"least-positive"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Num 1))))) ; theorem :: FVALUAT1:35 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k1_fvaluat1 :::"least-positive"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "v")) ")" )) "is" ($#v1_int_1 :::"integer"::: ) ))) ; theorem :: FVALUAT1:36 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool "ex" (Set (Var "i")) "being" ($#m1_hidden :::"Integer":::) "st" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "i")) ($#k4_xxreal_3 :::"*"::: ) (Set "(" ($#k1_fvaluat1 :::"least-positive"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "v")) ")" ) ")" ))))))) ; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "v" be ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Const "K")); assume (Bool (Set (Const "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) ; func :::"Pgenerator"::: "v" -> ($#m1_subset_1 :::"Element":::) "of" "K" equals :: FVALUAT1:def 9 "the" ($#m1_subset_1 :::"Element"::: ) "of" (Set "v" ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k1_fvaluat1 :::"least-positive"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) "v" ")" ) ")" ) ($#k1_tarski :::"}"::: ) )); end; :: deftheorem defines :::"Pgenerator"::: FVALUAT1:def 9 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k4_fvaluat1 :::"Pgenerator"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) "the" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Set (Var "v")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k1_fvaluat1 :::"least-positive"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "v")) ")" ) ")" ) ($#k1_tarski :::"}"::: ) ))))); definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "v" be ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Const "K")); assume (Bool (Set (Const "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) ; func :::"normal-valuation"::: "v" -> ($#m1_fvaluat1 :::"Valuation"::: ) "of" "K" means :: FVALUAT1:def 10 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" "K" "holds" (Bool (Set "v" ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" it ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) ($#k4_xxreal_3 :::"*"::: ) (Set "(" ($#k1_fvaluat1 :::"least-positive"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) "v" ")" ) ")" )))); end; :: deftheorem defines :::"normal-valuation"::: FVALUAT1:def 10 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")))) "iff" (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "b3")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) ($#k4_xxreal_3 :::"*"::: ) (Set "(" ($#k1_fvaluat1 :::"least-positive"::: ) (Set "(" ($#k2_relset_1 :::"rng"::: ) (Set (Var "v")) ")" ) ")" )))) ")" )))); theorem :: FVALUAT1:37 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) "iff" (Bool (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" )))) ; theorem :: FVALUAT1:38 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) "iff" (Bool (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k1_xxreal_0 :::"+infty"::: ) )) ")" )))) ; theorem :: FVALUAT1:39 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "," (Set (Var "b")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b")))) "iff" (Bool (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "b")))) ")" )))) ; theorem :: FVALUAT1:40 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) "is" ($#v2_xxreal_0 :::"positive"::: ) ) "iff" (Bool (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) "is" ($#v2_xxreal_0 :::"positive"::: ) ) ")" )))) ; theorem :: FVALUAT1:41 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")))) "iff" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "a")))) ")" )))) ; theorem :: FVALUAT1:42 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Bool "not" (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) "is" ($#v3_xxreal_0 :::"negative"::: ) )) "iff" (Bool "not" (Bool (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) "is" ($#v3_xxreal_0 :::"negative"::: ) )) ")" )))) ; theorem :: FVALUAT1:43 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" ($#k4_fvaluat1 :::"Pgenerator"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Num 1)))) ; theorem :: FVALUAT1:44 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))))) "holds" (Bool (Set (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))))))) ; theorem :: FVALUAT1:45 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Num 1))) "holds" (Bool (Set ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v"))) ($#r2_funct_2 :::"="::: ) (Set (Var "v")))))) ; theorem :: FVALUAT1:46 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" )) ($#r2_funct_2 :::"="::: ) (Set ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")))))) ; begin definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "v" be ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Const "K")); func :::"NonNegElements"::: "v" -> ($#m1_hidden :::"set"::: ) equals :: FVALUAT1:def 11 "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" "K" : (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set "v" ($#k3_funct_2 :::"."::: ) (Set (Var "x")))) "}" ; end; :: deftheorem defines :::"NonNegElements"::: FVALUAT1:def 11 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "holds" (Bool (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) : (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")))) "}" ))); theorem :: FVALUAT1:47 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")))) "iff" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")))) ")" )))) ; theorem :: FVALUAT1:48 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "holds" (Bool (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "K")))))) ; theorem :: FVALUAT1:49 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")))))) ; theorem :: FVALUAT1:50 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "K"))) ($#r2_hidden :::"in"::: ) (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")))))) ; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "v" be ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Const "K")); assume (Bool (Set (Const "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) ; func :::"ValuatRing"::: "v" -> ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v5_group_1 :::"commutative"::: ) ($#l6_algstr_0 :::"Ring":::) means :: FVALUAT1:def 12 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) "v")) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" "K") ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) "v" ")" ) "," (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) "v" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" "K") ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) "v" ")" ) "," (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) "v" ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) "K")) & (Bool (Set "the" ($#u3_struct_0 :::"OneF"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) "K")) ")" ); end; :: deftheorem defines :::"ValuatRing"::: FVALUAT1:def 12 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "b3")) "being" ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v36_algstr_0 :::"strict"::: ) ($#v5_group_1 :::"commutative"::: ) ($#l6_algstr_0 :::"Ring":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")))) & (Bool (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "K"))) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")) ")" ) "," (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "K"))) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")) ")" ) "," (Set "(" ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")) ")" ) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set "the" ($#u3_struct_0 :::"OneF"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "K")))) ")" ) ")" )))); theorem :: FVALUAT1:51 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "holds" (Bool (Set (Var "x")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")))))) ; theorem :: FVALUAT1:52 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")))) "iff" (Bool (Set (Var "a")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )) ")" )))) ; theorem :: FVALUAT1:53 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "S")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) "is" ($#m2_xxreal_2 :::"LowerBound"::: ) "of" (Set (Set (Var "v")) ($#k7_relset_1 :::".:"::: ) (Set (Var "S"))))))) ; theorem :: FVALUAT1:54 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "x1")) "," (Set (Var "y1")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "y1")))) "holds" (Bool (Set (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x1")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "y1")))))))) ; theorem :: FVALUAT1:55 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "x1")) "," (Set (Var "y1")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "x1"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Var "y1")))) "holds" (Bool (Set (Set (Var "x")) ($#k8_group_1 :::"*"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "x1")) ($#k8_group_1 :::"*"::: ) (Set (Var "y1")))))))) ; theorem :: FVALUAT1:56 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))) ; theorem :: FVALUAT1:57 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "K")))))) ; theorem :: FVALUAT1:58 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "holds" (Bool (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set (Var "y")))))))) ; theorem :: FVALUAT1:59 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v"))) "is" ($#v1_vectsp_2 :::"domRing-like"::: ) ))) ; theorem :: FVALUAT1:60 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "holds" (Bool (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "y")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "y")) "," (Set (Var "n")) ")" )))))) ; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "v" be ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Const "K")); assume (Bool (Set (Const "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) ; func :::"PosElements"::: "v" -> ($#m1_subset_1 :::"Ideal":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) "v" ")" ) equals :: FVALUAT1:def 13 "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" "K" : (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set "v" ($#k3_funct_2 :::"."::: ) (Set (Var "x")))) "}" ; end; :: deftheorem defines :::"PosElements"::: FVALUAT1:def 13 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k8_fvaluat1 :::"PosElements"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "x")) where x "is" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) : (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")))) "}" ))); notationlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "v" be ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Const "K")); synonym :::"vp"::: "v" for :::"PosElements"::: "v"; end; theorem :: FVALUAT1:61 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v")))) "iff" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")))) ")" )))) ; theorem :: FVALUAT1:62 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v")))))) ; theorem :: FVALUAT1:63 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "not" (Bool (Set ($#k5_struct_0 :::"1."::: ) (Set (Var "K"))) ($#r2_hidden :::"in"::: ) (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v"))))))) ; definitionlet "K" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) ; let "v" be ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Const "K")); let "S" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Const "K")); assume that (Bool (Set (Const "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) and (Bool (Set (Const "S")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Const "v")) ")" )) ; func :::"min"::: "(" "S" "," "v" ")" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) "v" ")" ) equals :: FVALUAT1:def 14 (Set (Set "(" "v" ($#k8_relset_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k7_supinf_2 :::"inf"::: ) (Set "(" "v" ($#k7_relset_1 :::".:"::: ) "S" ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k13_ideal_1 :::"/\"::: ) "S"); end; :: deftheorem defines :::"min"::: FVALUAT1:def 14 : (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "S")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ))) "holds" (Bool (Set ($#k9_fvaluat1 :::"min"::: ) "(" (Set (Var "S")) "," (Set (Var "v")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "v")) ($#k8_relset_1 :::"""::: ) (Set ($#k6_domain_1 :::"{"::: ) (Set "(" ($#k7_supinf_2 :::"inf"::: ) (Set "(" (Set (Var "v")) ($#k7_relset_1 :::".:"::: ) (Set (Var "S")) ")" ) ")" ) ($#k6_domain_1 :::"}"::: ) ) ")" ) ($#k13_ideal_1 :::"/\"::: ) (Set (Var "S"))))))); theorem :: FVALUAT1:64 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "S")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ))) "holds" (Bool (Set ($#k9_fvaluat1 :::"min"::: ) "(" (Set (Var "S")) "," (Set (Var "v")) ")" ) ($#r1_tarski :::"c="::: ) (Set (Var "S")))))) ; theorem :: FVALUAT1:65 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) (Bool "for" (Set (Var "S")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "S")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ))) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_fvaluat1 :::"min"::: ) "(" (Set (Var "S")) "," (Set (Var "v")) ")" )) "iff" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "S"))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "S")))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")))) ")" ) ")" ) ")" ))))) ; theorem :: FVALUAT1:66 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "st" (Bool (Bool (Set (Var "I")) "is" ($#m1_subset_1 :::"Ideal":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k9_fvaluat1 :::"min"::: ) "(" (Set (Var "I")) "," (Set (Var "v")) ")" ))) "holds" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set (Set ($#k6_domain_1 :::"{"::: ) (Set (Var "x")) ($#k6_domain_1 :::"}"::: ) ) ($#k7_ideal_1 :::"-Ideal"::: ) )))))) ; theorem :: FVALUAT1:67 (Bool "for" (Set (Var "R")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "I")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_ideal_1 :::"add-closed"::: ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "R")) "holds" (Bool (Set (Var "I")) "is" ($#m1_realset1 :::"Preserv"::: ) "of" (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "R")))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "P" be ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Const "R")); mode :::"Submodule"::: "of" "P" -> ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set ($#k4_vectsp_2 :::"RightModule"::: ) "R") means :: FVALUAT1:def 15 (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) "P"); end; :: deftheorem defines :::"Submodule"::: FVALUAT1:def 15 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "b3")) "being" ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set ($#k4_vectsp_2 :::"RightModule"::: ) (Set (Var "R"))) "holds" (Bool "(" (Bool (Set (Var "b3")) "is" ($#m2_fvaluat1 :::"Submodule"::: ) "of" (Set (Var "P"))) "iff" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "P"))) ")" )))); registrationlet "R" be ($#l6_algstr_0 :::"Ring":::); let "P" be ($#m1_subset_1 :::"RightIdeal":::) "of" (Set (Const "R")); cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v5_algstr_0 :::"left_add-cancelable"::: ) ($#v6_algstr_0 :::"right_add-cancelable"::: ) ($#v7_algstr_0 :::"add-cancelable"::: ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v2_vectsp_2 :::"strict"::: ) ($#v4_vectsp_2 :::"RightMod-like"::: ) bbbadV1_ALGSTR_1() bbbadV2_ALGSTR_1() bbbadV3_ALGSTR_1() bbbadV4_ALGSTR_1() for ($#m2_fvaluat1 :::"Submodule"::: ) "of" "P"; end; theorem :: FVALUAT1:68 (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "P")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "M")) "being" ($#m2_fvaluat1 :::"Submodule"::: ) "of" (Set (Var "P")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"BinOp":::) "of" (Set (Var "P")) (Bool "for" (Set (Var "z")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set (Var "P")) (Bool "for" (Set (Var "m")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Var "P")) "st" (Bool (Bool (Set (Var "a")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "R"))) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set (Var "P")) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set (Var "m")) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "R"))) ($#k2_partfun1 :::"|"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Var "P")) "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "R"))) ($#k2_zfmisc_1 :::":]"::: ) ))) & (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "R"))))) "holds" (Bool (Set ($#g1_vectsp_2 :::"RightModStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "M"))) "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "M"))) "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "M"))) "," (Set "the" ($#u1_vectsp_2 :::"rmult"::: ) "of" (Set (Var "M"))) "#)" ) ($#r1_hidden :::"="::: ) (Set ($#g1_vectsp_2 :::"RightModStr"::: ) "(#" (Set (Var "P")) "," (Set (Var "a")) "," (Set (Var "z")) "," (Set (Var "m")) "#)" )))))))) ; definitionlet "R" be ($#l6_algstr_0 :::"Ring":::); let "M1", "M2" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "h" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "M1")) "," (Set (Const "M2")); attr "h" is :::"scalar-linear"::: means :: FVALUAT1:def 16 (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" "M1" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element":::) "of" "R" "holds" (Bool (Set "h" ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" "h" ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "r")))))); end; :: deftheorem defines :::"scalar-linear"::: FVALUAT1:def 16 : (Bool "for" (Set (Var "R")) "being" ($#l6_algstr_0 :::"Ring":::) (Bool "for" (Set (Var "M1")) "," (Set (Var "M2")) "being" ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Var "R")) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "M1")) "," (Set (Var "M2")) "holds" (Bool "(" (Bool (Set (Var "h")) "is" ($#v4_fvaluat1 :::"scalar-linear"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "M1")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "R")) "holds" (Bool (Set (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_vectsp_2 :::"*"::: ) (Set (Var "r")))))) ")" )))); registrationlet "R" be ($#l6_algstr_0 :::"Ring":::); let "M1" be ($#l1_vectsp_2 :::"RightMod":::) "of" (Set (Const "R")); let "M2" be ($#m1_rmod_2 :::"Submodule"::: ) "of" (Set (Const "M1")); cluster (Set ($#k1_yellow_9 :::"incl"::: ) "(" "M2" "," "M1" ")" ) -> ($#v13_vectsp_1 :::"additive"::: ) ($#v4_fvaluat1 :::"scalar-linear"::: ) ; end; theorem :: FVALUAT1:69 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "b")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "b")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a")) ")" ) ($#k3_supinf_2 :::"+"::: ) (Set "(" (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "b")) ")" )))))) ; theorem :: FVALUAT1:70 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ))) "holds" (Bool (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "a")) "," (Set (Var "n")) ")" ) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )))))) ; theorem :: FVALUAT1:71 (Bool "for" (Set (Var "n")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K"))))) "holds" (Bool (Set (Set "(" ($#k4_group_1 :::"power"::: ) (Set (Var "K")) ")" ) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "n")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))))))) ; theorem :: FVALUAT1:72 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "(" (Bool (Set (Var "a")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )) & (Bool (Set (Set (Var "a")) ($#k11_algstr_0 :::"""::: ) ) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )) ")" )))) ; theorem :: FVALUAT1:73 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Var "a")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "K")))) & (Bool (Set (Var "a")) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )) & (Bool (Set (Set (Var "a")) ($#k11_algstr_0 :::"""::: ) ) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ))) "holds" (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; theorem :: FVALUAT1:74 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "K")) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) ) & (Bool (Set (Set (Var "v")) ($#k3_funct_2 :::"."::: ) (Set (Var "a"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "for" (Set (Var "I")) "being" ($#m1_subset_1 :::"Ideal":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "holds" (Bool "(" (Bool (Set (Var "a")) ($#r2_hidden :::"in"::: ) (Set (Var "I"))) "iff" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ))) ")" ))))) ; theorem :: FVALUAT1:75 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k4_fvaluat1 :::"Pgenerator"::: ) (Set (Var "v"))) "is" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" )))) ; theorem :: FVALUAT1:76 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v"))) "is" ($#v1_subset_1 :::"proper"::: ) ))) ; theorem :: FVALUAT1:77 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "st" (Bool (Bool (Bool "not" (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v")))))) "holds" (Bool (Set (Set (Var "v")) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; theorem :: FVALUAT1:78 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v"))) "is" ($#v2_ring_1 :::"prime"::: ) ))) ; theorem :: FVALUAT1:79 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "I")) "being" ($#v1_subset_1 :::"proper"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "holds" (Bool (Set (Var "I")) ($#r1_tarski :::"c="::: ) (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v"))))))) ; theorem :: FVALUAT1:80 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v"))) "is" ($#v4_ring_1 :::"maximal"::: ) ))) ; theorem :: FVALUAT1:81 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool "for" (Set (Var "I")) "being" ($#v4_ring_1 :::"maximal"::: ) ($#m1_subset_1 :::"Ideal":::) "of" (Set "(" ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")) ")" ) "holds" (Bool (Set (Var "I")) ($#r1_hidden :::"="::: ) (Set ($#k8_fvaluat1 :::"vp"::: ) (Set (Var "v"))))))) ; theorem :: FVALUAT1:82 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k6_fvaluat1 :::"NonNegElements"::: ) (Set (Var "v")))))) ; theorem :: FVALUAT1:83 (Bool "for" (Set (Var "K")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#~v6_struct_0 "non" ($#v6_struct_0 :::"degenerated"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_vectsp_1 :::"distributive"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v1_realset2 :::"Field-like"::: ) ($#l6_algstr_0 :::"doubleLoopStr"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_fvaluat1 :::"Valuation"::: ) "of" (Set (Var "K")) "st" (Bool (Bool (Set (Var "K")) "is" ($#v3_fvaluat1 :::"having_valuation"::: ) )) "holds" (Bool (Set ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set "(" ($#k5_fvaluat1 :::"normal-valuation"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k7_fvaluat1 :::"ValuatRing"::: ) (Set (Var "v")))))) ;