:: PDIFF_7 semantic presentation begin registrationlet "n" be ($#m1_hidden :::"Nat":::); let "p", "q" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); let "f", "g" be ($#v3_valued_0 :::"real-valued"::: ) ($#m1_hidden :::"FinSequence":::); identify ; end; registrationlet "r", "s" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; let "n" be ($#m1_hidden :::"Nat":::); let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); let "f" be ($#v3_valued_0 :::"real-valued"::: ) ($#m1_hidden :::"FinSequence":::); identify ; end; registrationlet "n" be ($#m1_hidden :::"Nat":::); let "p" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); let "f" be ($#v3_valued_0 :::"real-valued"::: ) ($#m1_hidden :::"FinSequence":::); identify ; end; registrationlet "n" be ($#m1_hidden :::"Nat":::); let "p", "q" be ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k15_euclid :::"TOP-REAL"::: ) (Set (Const "n")) ")" ); let "f", "g" be ($#v3_valued_0 :::"real-valued"::: ) ($#m1_hidden :::"FinSequence":::); identify ; end; theorem :: PDIFF_7:1 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "j")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "i"))))) ; theorem :: PDIFF_7:2 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "j")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )))) ; theorem :: PDIFF_7:3 (Bool "for" (Set (Var "j")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "j")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set "(" (Set (Var "j")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" )))) ; theorem :: PDIFF_7:4 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "implies" (Bool (Set (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "j")) ")" ) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ))) ")" & (Bool (Set (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "j")) ")" ) ($#k2_rfinseq :::"/^"::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set "(" (Set (Var "j")) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ))) ")" )) ; theorem :: PDIFF_7:5 (Bool "for" (Set (Var "i")) "," (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "xi")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j")))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "j")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "xi")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "xi")) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: PDIFF_7:6 (Bool "for" (Set (Var "m")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" ) ($#k8_euclid :::"-"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "r")) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) & (Bool (Set (Set (Var "x")) ($#k8_euclid :::"-"::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "r")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set (Var "r")) ")" ))) ")" )))) ; theorem :: PDIFF_7:7 (Bool "for" (Set (Var "m")) "," (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "p")) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ")" ))) & (Bool (Set (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "p")) ")" ))) ")" )))) ; theorem :: PDIFF_7:8 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "Z")) "is" ($#v3_nfcont_1 :::"open"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r7_pdiff_1 :::"is_partial_differentiable_on"::: ) (Set (Var "Z")) "," (Set (Var "i"))) "iff" (Bool "(" (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "f")) ($#r2_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "x")) "," (Set (Var "i"))) ")" ) ")" ) ")" ))))) ; theorem :: PDIFF_7:9 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set "(" (Set (Var "x")) ($#k9_binop_2 :::"+"::: ) (Set (Var "y")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set (Var "y")) ")" ")" )))))) ; theorem :: PDIFF_7:10 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k10_rvsum_1 :::"*"::: ) (Set "(" ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" )))))) ; theorem :: PDIFF_7:11 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"<>"::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "m"))))))) ; theorem :: PDIFF_7:12 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_finseq_7 :::"Replace"::: ) "(" (Set (Var "z")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set "(" (Set (Var "x")) ($#k10_binop_2 :::"-"::: ) (Set (Var "y")) ")" ) ")" )) & (Bool (Set (Set (Var "z")) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_finseq_7 :::"Replace"::: ) "(" (Set (Var "z")) "," (Set (Var "i")) "," (Set (Var "x")) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_finseq_7 :::"Replace"::: ) "(" (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) "," (Set (Var "i")) "," (Set "(" (Set (Var "y")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x")) ")" ) ")" )) ")" ))))) ; theorem :: PDIFF_7:13 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k9_binop_2 :::"+"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_euclid :::"+"::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" )))))) ; theorem :: PDIFF_7:14 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" )))))) ; theorem :: PDIFF_7:15 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k11_binop_2 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k9_euclid :::"*"::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))))) ; theorem :: PDIFF_7:16 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool (Set (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ))))))) ; theorem :: PDIFF_7:17 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "m"))))))) ; theorem :: PDIFF_7:18 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" )))) "holds" (Bool (Set (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"<>"::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" )))))) ; theorem :: PDIFF_7:19 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "z")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_euclid :::"-"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k10_binop_2 :::"-"::: ) (Set (Var "y")) ")" ))) & (Bool (Set (Set (Var "z")) ($#k8_euclid :::"-"::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "z")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "y")) ($#k10_binop_2 :::"-"::: ) (Set (Var "x")) ")" ))) ")" ))))) ; theorem :: PDIFF_7:20 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_pdiff_1 :::"Proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "z"))))) "holds" (Bool "(" (Bool (Set (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "z")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "x")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "y")) ")" ))) & (Bool (Set (Set (Var "z")) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "z")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set (Var "y")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "x")) ")" ))) ")" ))))) ; theorem :: PDIFF_7:21 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_ndiff_1 :::"is_differentiable_in"::: ) (Set (Var "x"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "x")) "," (Set (Var "i"))) & (Bool (Set ($#k9_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_ndiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ")" ) ($#k3_relat_1 :::"*"::: ) (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ))) ")" ))))) ; theorem :: PDIFF_7:22 (Bool "for" (Set (Var "n")) "," (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "g")) ($#r1_pdiff_1 :::"is_differentiable_in"::: ) (Set (Var "y"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "g")) ($#r4_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "y")) "," (Set (Var "i"))) & (Bool (Set ($#k12_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "g")) "," (Set (Var "y")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k8_pdiff_1 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "y")) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k7_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) ")" ) ")" ")" ) ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Num 1) ($#k12_finseq_1 :::"*>"::: ) ))) ")" ))))) ; definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Const "n"))); pred "f" :::"is_differentiable_in"::: "x" means :: PDIFF_7:def 1 (Bool (Set ($#k3_pdiff_1 :::"<>*"::: ) "f") ($#r1_pdiff_1 :::"is_differentiable_in"::: ) "x"); end; :: deftheorem defines :::"is_differentiable_in"::: PDIFF_7:def 1 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_pdiff_7 :::"is_differentiable_in"::: ) (Set (Var "x"))) "iff" (Bool (Set ($#k3_pdiff_1 :::"<>*"::: ) (Set (Var "f"))) ($#r1_pdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x"))) ")" )))); definitionlet "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "x" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Const "n"))); func :::"diff"::: "(" "f" "," "x" ")" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: PDIFF_7:def 2 (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k8_pdiff_1 :::"diff"::: ) "(" (Set "(" ($#k3_pdiff_1 :::"<>*"::: ) "f" ")" ) "," "x" ")" ")" )); end; :: deftheorem defines :::"diff"::: PDIFF_7:def 2 : (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool (Set ($#k1_pdiff_7 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 1) ")" ")" ) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k8_pdiff_1 :::"diff"::: ) "(" (Set "(" ($#k3_pdiff_1 :::"<>*"::: ) (Set (Var "f")) ")" ) "," (Set (Var "x")) ")" ")" )))))); theorem :: PDIFF_7:23 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "h")) ($#r1_pdiff_7 :::"is_differentiable_in"::: ) (Set (Var "y"))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "h")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "y")) "," (Set (Var "i"))) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "h")) "," (Set (Var "y")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" (Set (Var "h")) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "y")) ")" ")" ) ")" ) "," (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ")" )) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "h")) "," (Set (Var "y")) "," (Set (Var "i")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_7 :::"diff"::: ) "(" (Set (Var "h")) "," (Set (Var "y")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set "(" ($#k5_euclid :::"0*"::: ) (Set (Var "m")) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Num 1) ")" ))) ")" ))))) ; theorem :: PDIFF_7:24 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "," (Set (Var "u")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "w")))) & (Bool (Set (Var "u")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "v")) ($#k7_integr15 :::"+"::: ) (Set (Var "w"))))) "holds" (Bool (Set ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "v")) ")" ) ($#k7_euclid :::"+"::: ) (Set "(" ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "w")) ")" ))))) ; theorem :: PDIFF_7:25 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "w")) "," (Set (Var "u")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "u")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k9_integr15 :::"(#)"::: ) (Set (Var "w"))))) "holds" (Bool (Set ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set "(" ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "w")) ")" )))))) ; theorem :: PDIFF_7:26 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "," (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "g")) ")" ) ($#k23_binop_2 :::"+"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ))) ")" )) "holds" (Bool (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Num 1) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "h")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "g")))))) ; theorem :: PDIFF_7:27 (Bool "for" (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "," (Set (Var "g")) "," (Set (Var "j")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "j")))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "j")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "j"))))) "holds" (Bool (Set (Set (Var "j")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ))) ")" )) "holds" (Bool (Set ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "j"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "h")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "g")) ")" ))))) ; theorem :: PDIFF_7:28 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "ex" (Set (Var "h")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m")))(Bool "ex" (Set (Var "g")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "st" (Bool "(" (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k23_binop_2 :::"+"::: ) (Num 1))) & (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "g"))) ($#r1_hidden :::"="::: ) (Set (Var "m"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set "(" (Set (Var "m")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" ($#k5_euclid :::"0*"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "g"))))) "holds" (Bool (Set (Set (Var "g")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) ")" ) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ")" ))) ")" ) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) (Bool "for" (Set (Var "hi")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "h")))) & (Bool (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Var "hi")))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set (Var "hi")) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k12_euclid :::"|."::: ) (Set (Var "y")) ($#k12_euclid :::".|"::: ) ))) ")" ) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set (Var "y")) ")" ) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "g")))) ")" )))))) ; theorem :: PDIFF_7:29 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 1) ")" ) (Bool "ex" (Set (Var "f0")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Set (Var "f")) ($#r2_relset_1 :::"="::: ) (Set ($#k3_pdiff_1 :::"<>*"::: ) (Set (Var "f0"))))))) ; theorem :: PDIFF_7:30 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "f0")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "x0")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "x0"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "f0")))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f0")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0"))))))))) ; definitionlet "m" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "X" be ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "m")) ")" ); attr "X" is :::"open"::: means :: PDIFF_7:def 3 (Bool "ex" (Set (Var "X0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "m" ")" ) "st" (Bool "(" (Bool (Set (Var "X0")) ($#r1_hidden :::"="::: ) "X") & (Bool (Set (Var "X0")) "is" ($#v3_nfcont_1 :::"open"::: ) ) ")" )); end; :: deftheorem defines :::"open"::: PDIFF_7:def 3 : (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_pdiff_7 :::"open"::: ) ) "iff" (Bool "ex" (Set (Var "X0")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool "(" (Bool (Set (Var "X0")) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool (Set (Var "X0")) "is" ($#v3_nfcont_1 :::"open"::: ) ) ")" )) ")" ))); theorem :: PDIFF_7:31 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "holds" (Bool "(" (Bool (Set (Var "X")) "is" ($#v1_pdiff_7 :::"open"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool "{" (Set (Var "y")) where y "is" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) : (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "y")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) "}" ($#r1_tarski :::"c="::: ) (Set (Var "X"))) ")" ))) ")" ))) ; definitionlet "m", "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_partial_differentiable_on"::: "X" "," "i" means :: PDIFF_7:def 4 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "m") "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "X") ($#r4_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "x")) "," "i") ")" ) ")" ); end; :: deftheorem defines :::"is_partial_differentiable_on"::: PDIFF_7:def 4 : (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_7 :::"is_partial_differentiable_on"::: ) (Set (Var "X")) "," (Set (Var "i"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r4_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "x")) "," (Set (Var "i"))) ")" ) ")" ) ")" ))))); theorem :: PDIFF_7:32 (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_7 :::"is_partial_differentiable_on"::: ) (Set (Var "X")) "," (Set (Var "i")))) "holds" (Bool (Set (Var "X")) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" )))))) ; theorem :: PDIFF_7:33 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_7 :::"is_partial_differentiable_on"::: ) (Set (Var "Z")) "," (Set (Var "i"))) "iff" (Bool (Set (Var "g")) ($#r7_pdiff_1 :::"is_partial_differentiable_on"::: ) (Set (Var "Z")) "," (Set (Var "i"))) ")" )))))) ; theorem :: PDIFF_7:34 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "Z")) "is" ($#v1_pdiff_7 :::"open"::: ) ) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_7 :::"is_partial_differentiable_on"::: ) (Set (Var "Z")) "," (Set (Var "i"))) "iff" (Bool "(" (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "f")) ($#r4_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "x")) "," (Set (Var "i"))) ")" ) ")" ) ")" ))))) ; definitionlet "m", "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "i" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "X" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r2_pdiff_7 :::"is_partial_differentiable_on"::: ) (Set (Const "X")) "," (Set (Const "i"))) ; func "f" :::"`partial|"::: "(" "X" "," "i" ")" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) "m" ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) "n" ")" ) means :: PDIFF_7:def 5 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "X") & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "m") "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k12_pdiff_1 :::"partdiff"::: ) "(" "f" "," (Set (Var "x")) "," "i" ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`partial|"::: PDIFF_7:def 5 : (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_7 :::"is_partial_differentiable_on"::: ) (Set (Var "X")) "," (Set (Var "i")))) "holds" (Bool "for" (Set (Var "b6")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set (Var "b6")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_pdiff_7 :::"`partial|"::: ) "(" (Set (Var "X")) "," (Set (Var "i")) ")" )) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b6"))) ($#r1_hidden :::"="::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "b6")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k12_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) "," (Set (Var "i")) ")" )) ")" ) ")" ) ")" )))))); definitionlet "m", "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "x0" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Const "m"))); pred "f" :::"is_continuous_in"::: "x0" means :: PDIFF_7:def 6 (Bool "ex" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "m" ")" )(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "m" ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) "n" ")" ) "st" (Bool "(" (Bool "x0" ($#r1_hidden :::"="::: ) (Set (Var "y0"))) & (Bool "f" ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "y0"))) ")" ))); end; :: deftheorem defines :::"is_continuous_in"::: PDIFF_7:def 6 : (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_pdiff_7 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "ex" (Set (Var "y0")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" )(Bool "ex" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) "st" (Bool "(" (Bool (Set (Var "x0")) ($#r1_hidden :::"="::: ) (Set (Var "y0"))) & (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "g")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "y0"))) ")" ))) ")" )))); theorem :: PDIFF_7:35 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g"))) & (Bool (Set (Var "x")) ($#r1_hidden :::"="::: ) (Set (Var "y")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_pdiff_7 :::"is_continuous_in"::: ) (Set (Var "x"))) "iff" (Bool (Set (Var "g")) ($#r1_nfcont_1 :::"is_continuous_in"::: ) (Set (Var "y"))) ")" )))))) ; theorem :: PDIFF_7:36 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "x0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_pdiff_7 :::"is_continuous_in"::: ) (Set (Var "x0"))) "iff" (Bool "(" (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "x1")) ($#k8_euclid :::"-"::: ) (Set (Var "x0")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" )) ")" ) ")" ) ")" )))) ; definitionlet "m", "n" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ); let "X" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_continuous_on"::: "X" means :: PDIFF_7:def 7 (Bool "(" (Bool "X" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "m") "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) "X")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "X") ($#r3_pdiff_7 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ); end; :: deftheorem defines :::"is_continuous_on"::: PDIFF_7:def 7 : (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_pdiff_7 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r3_pdiff_7 :::"is_continuous_in"::: ) (Set (Var "x0"))) ")" ) ")" ) ")" )))); theorem :: PDIFF_7:37 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_pdiff_7 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool (Set (Var "g")) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" ))))) ; theorem :: PDIFF_7:38 (Bool "for" (Set (Var "m")) "," (Set (Var "n")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_pdiff_7 :::"is_continuous_on"::: ) (Set (Var "X"))) "iff" (Bool "(" (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x0")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r")))) "holds" (Bool "ex" (Set (Var "s")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s"))) & (Bool "(" "for" (Set (Var "x1")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x1")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "x1")) ($#k8_euclid :::"-"::: ) (Set (Var "x0")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "s")))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x1")) ")" ) ($#k8_euclid :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x0")) ")" ) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "r"))) ")" ) ")" ))) ")" ) ")" ) ")" )))) ; theorem :: PDIFF_7:39 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "xi")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "xi"))))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Var "xi"))))))) ; theorem :: PDIFF_7:40 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "xi")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "xi"))))) "holds" (Bool (Set ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ) ($#r2_funct_2 :::"="::: ) (Set ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "y")) ")" ))))))) ; theorem :: PDIFF_7:41 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "xi")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "xi")))) & (Bool (Set (Var "g")) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" )))) "holds" (Bool (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set (Var "g")) "," (Set (Var "xi")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "y")) "," (Set (Var "i")) ")" )))))))) ; theorem :: PDIFF_7:42 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) & (Bool "(" "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "h"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ) & (Bool "(" "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "h"))) "," (Set (Var "i"))) ")" )) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k2_rcomp_1 :::"]."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k2_rcomp_1 :::".["::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "r")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "q")) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "y")) "," (Set (Var "i")) ")" ")" ))) ")" )))))))) ; theorem :: PDIFF_7:43 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "p")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "q"))) & (Bool "(" "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "h"))) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) ")" ) & (Bool "(" "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "h")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_rcomp_1 :::".]"::: ) ))) "holds" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "h"))) "," (Set (Var "i"))) ")" )) "holds" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::)(Bool "ex" (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool "(" (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set (Var "y")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "r")))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q")) ")" ) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "x")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "q")) ($#k10_binop_2 :::"-"::: ) (Set (Var "p")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "y")) "," (Set (Var "i")) ")" ")" ))) ")" )))))))) ; theorem :: PDIFF_7:44 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "," (Set (Var "w")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "d")) "," (Set (Var "p")) "," (Set (Var "q")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "y")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "z")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "p")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")))) & (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "y")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "q")))) & (Bool (Set (Var "r")) ($#r2_hidden :::"in"::: ) (Set ($#k1_rcomp_1 :::"[."::: ) (Set (Var "p")) "," (Set (Var "q")) ($#k1_rcomp_1 :::".]"::: ) )) & (Bool (Set (Var "w")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "y")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "r"))))) "holds" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "w")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))))))) ; theorem :: PDIFF_7:45 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "," (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "d")) "," (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool (Set (Var "X")) "is" ($#v1_pdiff_7 :::"open"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "y")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "z")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "X")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "x")) "," (Set (Var "i"))) ")" ) & (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "z")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d")))) "holds" (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) ")" ) & (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "y")) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set (Var "p")))) & (Bool (Set (Var "q")) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y"))))) "holds" (Bool "ex" (Set (Var "w")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool "(" (Bool (Set ($#k12_euclid :::"|."::: ) (Set "(" (Set (Var "w")) ($#k8_euclid :::"-"::: ) (Set (Var "x")) ")" ) ($#k12_euclid :::".|"::: ) ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "d"))) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "w")) "," (Set (Var "i"))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "z")) ")" ) ($#k10_binop_2 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "y")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "p")) ($#k10_binop_2 :::"-"::: ) (Set (Var "q")) ")" ) ($#k11_binop_2 :::"*"::: ) (Set "(" ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "w")) "," (Set (Var "i")) ")" ")" ))) ")" )))))))) ; theorem :: PDIFF_7:46 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "h")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "y")) "," (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "for" (Set (Var "j")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "m")) ($#k23_binop_2 :::"+"::: ) (Num 1))) & (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "j"))) & (Bool (Set (Var "j")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_hidden :::"Nat":::) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "h"))))) "holds" (Bool (Set (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "y")) ($#k17_finseq_1 :::"|"::: ) (Set "(" (Set "(" (Set (Var "m")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "i")) ")" ) ")" ) ($#k8_finseq_1 :::"^"::: ) (Set "(" ($#k5_euclid :::"0*"::: ) (Set "(" (Set (Var "i")) ($#k7_nat_d :::"-'"::: ) (Num 1) ")" ) ")" ))) ")" )) "holds" (Bool (Set (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set (Var "j")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set "(" (Set "(" (Set (Var "m")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "j")) ")" ) "," (Set "(" (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set "(" (Set (Var "h")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "j")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ")" ")" ) ($#k3_funct_2 :::"."::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set "(" (Set "(" (Set (Var "m")) ($#k23_binop_2 :::"+"::: ) (Num 1) ")" ) ($#k7_nat_d :::"-'"::: ) (Set (Var "j")) ")" ) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set (Var "y")) ")" ) ")" ))))))) ; theorem :: PDIFF_7:47 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) "st" (Bool (Bool (Set (Var "X")) "is" ($#v1_pdiff_7 :::"open"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_7 :::"is_partial_differentiable_on"::: ) (Set (Var "X")) "," (Set (Var "i"))) & (Bool (Set (Set (Var "f")) ($#k2_pdiff_7 :::"`partial|"::: ) "(" (Set (Var "X")) "," (Set (Var "i")) ")" ) ($#r4_pdiff_7 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_pdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x"))) & (Bool "(" "for" (Set (Var "h")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "m"))) (Bool "ex" (Set (Var "w")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 1)) "st" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set (Var "w")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "h")) ")" ) ($#k9_euclid :::"*"::: ) (Set "(" ($#k12_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) "," (Set (Var "i")) ")" ")" ))) ")" ) & (Bool (Set (Set "(" ($#k8_pdiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ")" ) ($#k1_pdiff_6 :::"."::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "w")))) ")" )) ")" ) ")" ))))) ; theorem :: PDIFF_7:48 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v3_nfcont_1 :::"open"::: ) ) & (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "X"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r7_pdiff_1 :::"is_partial_differentiable_on"::: ) (Set (Var "X")) "," (Set (Var "i"))) & (Bool (Set (Set (Var "f")) ($#k15_pdiff_1 :::"`partial|"::: ) "(" (Set (Var "X")) "," (Set (Var "i")) ")" ) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" ) ")" )) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_ndiff_1 :::"is_differentiable_in"::: ) (Set (Var "x"))) & (Bool "(" "for" (Set (Var "h")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) (Bool "ex" (Set (Var "w")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 1)) "st" (Bool "(" (Bool (Set ($#k4_finseq_1 :::"dom"::: ) (Set (Var "w"))) ($#r1_hidden :::"="::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m")))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Set (Var "i")) ($#r2_hidden :::"in"::: ) (Set ($#k2_finseq_1 :::"Seg"::: ) (Set (Var "m"))))) "holds" (Bool (Set (Set (Var "w")) ($#k1_funct_1 :::"."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k9_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) "," (Set (Var "i")) ")" ")" ) ($#k1_funct_1 :::"."::: ) (Set ($#k12_finseq_1 :::"<*"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Set (Var "i")) "," (Set (Var "m")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "h")) ")" ) ($#k12_finseq_1 :::"*>"::: ) ))) ")" ) & (Bool (Set (Set "(" ($#k3_ndiff_1 :::"diff"::: ) "(" (Set (Var "f")) "," (Set (Var "x")) ")" ")" ) ($#k17_lopban_1 :::"."::: ) (Set (Var "h"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid_7 :::"Sum"::: ) (Set (Var "w")))) ")" )) ")" ) ")" ))))) ; theorem :: PDIFF_7:49 (Bool "for" (Set (Var "m")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "," (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Num 1) ")" ) (Bool "for" (Set (Var "X")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k4_real_ns1 :::"REAL-NS"::: ) (Set (Var "m")) ")" ) "st" (Bool (Bool (Set (Var "X")) "is" ($#v3_nfcont_1 :::"open"::: ) )) "holds" (Bool "(" (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "m")))) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r7_pdiff_1 :::"is_partial_differentiable_on"::: ) (Set (Var "X")) "," (Set (Var "i"))) & (Bool (Set (Set (Var "f")) ($#k15_pdiff_1 :::"`partial|"::: ) "(" (Set (Var "X")) "," (Set (Var "i")) ")" ) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" ) ")" ) "iff" (Bool "(" (Bool (Set (Var "f")) ($#r2_ndiff_1 :::"is_differentiable_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f")) ($#k4_ndiff_1 :::"`|"::: ) (Set (Var "X"))) ($#r3_nfcont_1 :::"is_continuous_on"::: ) (Set (Var "X"))) ")" ) ")" )))) ;