:: PDIFF_2 semantic presentation begin theorem :: PDIFF_2:1 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 2) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_euclid :::"REAL"::: ) (Num 2))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 2) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "x"))) ")" ) ")" ) ; theorem :: PDIFF_2:2 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 2) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_euclid :::"REAL"::: ) (Num 2))) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 2) ")" ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_numbers :::"REAL"::: ) )) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) ")" ) ")" ) ; begin definitionlet "n", "i" be ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Const "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "z" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Const "n"))); func :::"SVF1"::: "(" "i" "," "f" "," "z" ")" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: PDIFF_2:def 1 (Set "f" ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" "i" "," "z" ")" ")" )); end; :: deftheorem defines :::"SVF1"::: PDIFF_2:def 1 : (Bool "for" (Set (Var "n")) "," (Set (Var "i")) "being" ($#m2_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Set (Var "n")) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Set (Var "i")) "," (Set (Var "f")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set "(" ($#k6_pdiff_1 :::"reproj"::: ) "(" (Set (Var "i")) "," (Set (Var "z")) ")" ")" )))))); theorem :: PDIFF_2:3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x")))))) ; theorem :: PDIFF_2:4 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x")) "," (Set (Var "y")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "y")))))) ; theorem :: PDIFF_2:5 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "holds" (Bool "(" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0"))) ")" )) "iff" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1)) ")" ))) ; theorem :: PDIFF_2:6 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "holds" (Bool "(" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "y0"))) ")" )) "iff" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2)) ")" ))) ; theorem :: PDIFF_2:7 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1))) "holds" (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" ))))) ; theorem :: PDIFF_2:8 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2))) "holds" (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "y")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "y")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" ))))) ; theorem :: PDIFF_2:9 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1)) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))) ; theorem :: PDIFF_2:10 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2)) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "y")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "y")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))) ; theorem :: PDIFF_2:11 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1))) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z")) "," (Num 1) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "x")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))) ; theorem :: PDIFF_2:12 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2))) "holds" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z")) "," (Num 2) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool (Set (Var "r")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "y")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "y")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "y")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "y")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))) ; theorem :: PDIFF_2:13 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1))) "holds" (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) "," (Set (Var "x0")) ")" ))))) ; theorem :: PDIFF_2:14 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"="::: ) (Set ($#k10_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) ($#k10_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2))) "holds" (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z")) ")" ")" ) "," (Set (Var "y0")) ")" ))))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "Z" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_partial_differentiable`1_on"::: "Z" means :: PDIFF_2:def 2 (Bool "(" (Bool "Z" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) "Z")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "Z") ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1)) ")" ) ")" ); pred "f" :::"is_partial_differentiable`2_on"::: "Z" means :: PDIFF_2:def 3 (Bool "(" (Bool "Z" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) "Z")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "Z") ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2)) ")" ) ")" ); end; :: deftheorem defines :::"is_partial_differentiable`1_on"::: PDIFF_2:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_pdiff_2 :::"is_partial_differentiable`1_on"::: ) (Set (Var "Z"))) "iff" (Bool "(" (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Z"))) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1)) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_partial_differentiable`2_on"::: PDIFF_2:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_2 :::"is_partial_differentiable`2_on"::: ) (Set (Var "Z"))) "iff" (Bool "(" (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Z"))) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2)) ")" ) ")" ) ")" ))); theorem :: PDIFF_2:15 (Bool "for" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_pdiff_2 :::"is_partial_differentiable`1_on"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 1)) ")" ) ")" ))) ; theorem :: PDIFF_2:16 (Bool "for" (Set (Var "Z")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_2 :::"is_partial_differentiable`2_on"::: ) (Set (Var "Z")))) "holds" (Bool "(" (Bool (Set (Var "Z")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z")) "," (Num 2)) ")" ) ")" ))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "Z" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r1_pdiff_2 :::"is_partial_differentiable`1_on"::: ) (Set (Const "Z"))) ; func "f" :::"`partial1|"::: "Z" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_2:def 4 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "Z") & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) "Z")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" "f" "," (Set (Var "z")) "," (Num 1) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`partial1|"::: PDIFF_2:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_pdiff_2 :::"is_partial_differentiable`1_on"::: ) (Set (Var "Z")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_pdiff_2 :::"`partial1|"::: ) (Set (Var "Z")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z")) "," (Num 1) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "Z" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r2_pdiff_2 :::"is_partial_differentiable`2_on"::: ) (Set (Const "Z"))) ; func "f" :::"`partial2|"::: "Z" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_2:def 5 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "Z") & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) "Z")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" "f" "," (Set (Var "z")) "," (Num 2) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`partial2|"::: PDIFF_2:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "Z")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_2 :::"is_partial_differentiable`2_on"::: ) (Set (Var "Z")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_pdiff_2 :::"`partial2|"::: ) (Set (Var "Z")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "Z"))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "Z")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "z"))) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z")) "," (Num 2) ")" )) ")" ) ")" ) ")" )))); begin theorem :: PDIFF_2:17 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0"))) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 1)) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "c")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_2:18 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0"))) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 2)) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#v2_relat_1 :::"non-zero"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#v1_fdiff_1 :::"-convergent"::: ) ($#m1_subset_1 :::"Real_Sequence":::) (Bool "for" (Set (Var "c")) "being" ($#v3_funct_1 :::"constant"::: ) ($#m1_subset_1 :::"Real_Sequence":::) "st" (Bool (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Var "N")))) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set (Var "f")) "," (Set (Var "z0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_2:19 (Bool "for" (Set (Var "z0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 1)) & (Bool (Set (Var "f2")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 1))) "holds" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 1)))) ; theorem :: PDIFF_2:20 (Bool "for" (Set (Var "z0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 2)) & (Bool (Set (Var "f2")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 2))) "holds" (Bool (Set (Set (Var "f1")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "f2"))) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 2)))) ; theorem :: PDIFF_2:21 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 1))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")))))) ; theorem :: PDIFF_2:22 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 2) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "z0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 2)) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "z0")) "," (Num 2))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set (Var "f")) "," (Set (Var "z0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 2) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")))))) ;