:: PDIFF_5 semantic presentation begin definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); pred "f" :::"is_hpartial_differentiable`11_in"::: "u" means :: PDIFF_5:def 1 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`12_in"::: "u" means :: PDIFF_5:def 2 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`13_in"::: "u" means :: PDIFF_5:def 3 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ))) & (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 "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`21_in"::: "u" means :: PDIFF_5:def 4 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`22_in"::: "u" means :: PDIFF_5:def 5 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`23_in"::: "u" means :: PDIFF_5:def 6 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ))) & (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 "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`31_in"::: "u" means :: PDIFF_5:def 7 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`32_in"::: "u" means :: PDIFF_5:def 8 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" )) ")" )); pred "f" :::"is_hpartial_differentiable`33_in"::: "u" means :: PDIFF_5:def 9 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ))) & (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 "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" )))))) ")" )) ")" )); end; :: deftheorem defines :::"is_hpartial_differentiable`11_in"::: PDIFF_5:def 1 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`12_in"::: PDIFF_5:def 2 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`13_in"::: PDIFF_5:def 3 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`21_in"::: PDIFF_5:def 4 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`22_in"::: PDIFF_5:def 5 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`23_in"::: PDIFF_5:def 6 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`31_in"::: PDIFF_5:def 7 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`32_in"::: PDIFF_5:def 8 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`33_in"::: PDIFF_5:def 9 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u"))) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" )))))) ")" )) ")" )) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff11"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 10 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#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")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff11"::: PDIFF_5:def 10 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff12"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 11 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#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")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff12"::: PDIFF_5:def 11 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff13"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 12 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 1) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff13"::: PDIFF_5:def 12 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff21"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 13 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#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")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff21"::: PDIFF_5:def 13 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff22"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 14 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#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")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff22"::: PDIFF_5:def 14 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff23"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 15 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 2) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff23"::: PDIFF_5:def 15 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff31"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 16 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#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")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff31"::: PDIFF_5:def 16 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k7_pdiff_5 :::"hpartdiff31"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "x0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "x0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "x0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff32"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 17 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#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 "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#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")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff32"::: PDIFF_5:def 17 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k8_pdiff_5 :::"hpartdiff32"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "y0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "y0")) ")" )) ($#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")) ($#k9_real_1 :::"-"::: ) (Set (Var "y0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "u" be ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)); assume (Bool (Set (Const "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Const "u"))) ; func :::"hpartdiff33"::: "(" "f" "," "u" ")" -> ($#m1_subset_1 :::"Real":::) means :: PDIFF_5:def 18 (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool "u" ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ))) & (Bool "ex" (Set (Var "L")) "being" ($#m1_subset_1 :::"LinearFunc":::)(Bool "ex" (Set (Var "R")) "being" ($#m1_subset_1 :::"RestFunc":::) "st" (Bool "(" (Bool it ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" "f" "," (Num 3) ")" ")" ) "," "u" ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )); end; :: deftheorem defines :::"hpartdiff33"::: PDIFF_5:def 18 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Real":::) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k9_pdiff_5 :::"hpartdiff33"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) "iff" (Bool "ex" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "(" (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool "ex" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Var "z0")) "st" (Bool "(" (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ))) & (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 "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Num 1))) & (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "z")) ($#r2_hidden :::"in"::: ) (Set (Var "N")))) "holds" (Bool (Set (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "z0")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "L")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "R")) ($#k1_seq_1 :::"."::: ) (Set "(" (Set (Var "z")) ($#k9_real_1 :::"-"::: ) (Set (Var "z0")) ")" ) ")" ))) ")" ) ")" ))) ")" )) ")" )) ")" )))); theorem :: PDIFF_5:1 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))))) ; theorem :: PDIFF_5:2 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "y0")))))) ; theorem :: PDIFF_5:3 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "z0")))))) ; theorem :: PDIFF_5:4 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))))) ; theorem :: PDIFF_5:5 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "y0")))))) ; theorem :: PDIFF_5:6 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "z0")))))) ; theorem :: PDIFF_5:7 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "x0")))))) ; theorem :: PDIFF_5:8 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "y0")))))) ; theorem :: PDIFF_5:9 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ) ($#r1_fdiff_1 :::"is_differentiable_in"::: ) (Set (Var "z0")))))) ; theorem :: PDIFF_5:10 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "x0")) ")" ))))) ; theorem :: PDIFF_5:11 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "y0")) ")" ))))) ; theorem :: PDIFF_5:12 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "z0")) ")" ))))) ; theorem :: PDIFF_5:13 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "x0")) ")" ))))) ; theorem :: PDIFF_5:14 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "y0")) ")" ))))) ; theorem :: PDIFF_5:15 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "z0")) ")" ))))) ; theorem :: PDIFF_5:16 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k7_pdiff_5 :::"hpartdiff31"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "x0")) ")" ))))) ; theorem :: PDIFF_5:17 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k8_pdiff_5 :::"hpartdiff32"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "y0")) ")" ))))) ; theorem :: PDIFF_5:18 (Bool "for" (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "u")) ($#r1_hidden :::"="::: ) (Set ($#k11_finseq_1 :::"<*"::: ) (Set (Var "x0")) "," (Set (Var "y0")) "," (Set (Var "z0")) ($#k11_finseq_1 :::"*>"::: ) )) & (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k9_pdiff_5 :::"hpartdiff33"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k1_fdiff_1 :::"diff"::: ) "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) ")" ")" ) "," (Set (Var "z0")) ")" ))))) ; definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_hpartial_differentiable`11_on"::: "D" means :: PDIFF_5:def 19 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`12_on"::: "D" means :: PDIFF_5:def 20 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`13_on"::: "D" means :: PDIFF_5:def 21 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`21_on"::: "D" means :: PDIFF_5:def 22 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`22_on"::: "D" means :: PDIFF_5:def 23 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`23_on"::: "D" means :: PDIFF_5:def 24 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`31_on"::: "D" means :: PDIFF_5:def 25 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`32_on"::: "D" means :: PDIFF_5:def 26 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u"))) ")" ) ")" ); pred "f" :::"is_hpartial_differentiable`33_on"::: "D" means :: PDIFF_5:def 27 (Bool "(" (Bool "D" ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set "f" ($#k2_partfun1 :::"|"::: ) "D") ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u"))) ")" ) ")" ); end; :: deftheorem defines :::"is_hpartial_differentiable`11_on"::: PDIFF_5:def 19 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r10_pdiff_5 :::"is_hpartial_differentiable`11_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`12_on"::: PDIFF_5:def 20 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r11_pdiff_5 :::"is_hpartial_differentiable`12_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`13_on"::: PDIFF_5:def 21 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r12_pdiff_5 :::"is_hpartial_differentiable`13_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`21_on"::: PDIFF_5:def 22 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r13_pdiff_5 :::"is_hpartial_differentiable`21_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`22_on"::: PDIFF_5:def 23 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r14_pdiff_5 :::"is_hpartial_differentiable`22_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`23_on"::: PDIFF_5:def 24 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r15_pdiff_5 :::"is_hpartial_differentiable`23_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`31_on"::: PDIFF_5:def 25 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r16_pdiff_5 :::"is_hpartial_differentiable`31_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`32_on"::: PDIFF_5:def 26 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r17_pdiff_5 :::"is_hpartial_differentiable`32_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); :: deftheorem defines :::"is_hpartial_differentiable`33_on"::: PDIFF_5:def 27 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r18_pdiff_5 :::"is_hpartial_differentiable`33_on"::: ) (Set (Var "D"))) "iff" (Bool "(" (Bool (Set (Var "D")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "D"))) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u"))) ")" ) ")" ) ")" ))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r10_pdiff_5 :::"is_hpartial_differentiable`11_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial11|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 28 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial11|"::: PDIFF_5:def 28 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r10_pdiff_5 :::"is_hpartial_differentiable`11_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k10_pdiff_5 :::"`hpartial11|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r11_pdiff_5 :::"is_hpartial_differentiable`12_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial12|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 29 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial12|"::: PDIFF_5:def 29 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r11_pdiff_5 :::"is_hpartial_differentiable`12_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k11_pdiff_5 :::"`hpartial12|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r12_pdiff_5 :::"is_hpartial_differentiable`13_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial13|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 30 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial13|"::: PDIFF_5:def 30 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r12_pdiff_5 :::"is_hpartial_differentiable`13_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k12_pdiff_5 :::"`hpartial13|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r13_pdiff_5 :::"is_hpartial_differentiable`21_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial21|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 31 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial21|"::: PDIFF_5:def 31 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r13_pdiff_5 :::"is_hpartial_differentiable`21_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k13_pdiff_5 :::"`hpartial21|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r14_pdiff_5 :::"is_hpartial_differentiable`22_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial22|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 32 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial22|"::: PDIFF_5:def 32 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r14_pdiff_5 :::"is_hpartial_differentiable`22_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k14_pdiff_5 :::"`hpartial22|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r15_pdiff_5 :::"is_hpartial_differentiable`23_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial23|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 33 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial23|"::: PDIFF_5:def 33 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r15_pdiff_5 :::"is_hpartial_differentiable`23_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k15_pdiff_5 :::"`hpartial23|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r16_pdiff_5 :::"is_hpartial_differentiable`31_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial31|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 34 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k7_pdiff_5 :::"hpartdiff31"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial31|"::: PDIFF_5:def 34 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r16_pdiff_5 :::"is_hpartial_differentiable`31_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k16_pdiff_5 :::"`hpartial31|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k7_pdiff_5 :::"hpartdiff31"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r17_pdiff_5 :::"is_hpartial_differentiable`32_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial32|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 35 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k8_pdiff_5 :::"hpartdiff32"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial32|"::: PDIFF_5:def 35 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r17_pdiff_5 :::"is_hpartial_differentiable`32_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k17_pdiff_5 :::"`hpartial32|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k8_pdiff_5 :::"hpartdiff32"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); definitionlet "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "D" be ($#m1_hidden :::"set"::: ) ; assume (Bool (Set (Const "f")) ($#r18_pdiff_5 :::"is_hpartial_differentiable`33_on"::: ) (Set (Const "D"))) ; func "f" :::"`hpartial33|"::: "D" -> ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) means :: PDIFF_5:def 36 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) "D") & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) "D")) "holds" (Bool (Set it ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k9_pdiff_5 :::"hpartdiff33"::: ) "(" "f" "," (Set (Var "u")) ")" )) ")" ) ")" ); end; :: deftheorem defines :::"`hpartial33|"::: PDIFF_5:def 36 : (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "D")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r18_pdiff_5 :::"is_hpartial_differentiable`33_on"::: ) (Set (Var "D")))) "holds" (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k18_pdiff_5 :::"`hpartial33|"::: ) (Set (Var "D")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Var "D"))) & (Bool "(" "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "u")) ($#r2_hidden :::"in"::: ) (Set (Var "D")))) "holds" (Bool (Set (Set (Var "b3")) ($#k1_seq_1 :::"."::: ) (Set (Var "u"))) ($#r1_hidden :::"="::: ) (Set ($#k9_pdiff_5 :::"hpartdiff33"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" )) ")" ) ")" ) ")" )))); begin theorem :: PDIFF_5:19 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 1)) ")" ))) ; theorem :: PDIFF_5:20 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 2)) ")" ))) ; theorem :: PDIFF_5:21 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 3)) ")" ))) ; theorem :: PDIFF_5:22 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 1)) ")" ))) ; theorem :: PDIFF_5:23 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 2)) ")" ))) ; theorem :: PDIFF_5:24 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 3)) ")" ))) ; theorem :: PDIFF_5:25 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 1)) ")" ))) ; theorem :: PDIFF_5:26 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 2)) ")" ))) ; theorem :: PDIFF_5:27 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u"))) "iff" (Bool (Set ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u")) "," (Num 3)) ")" ))) ; theorem :: PDIFF_5:28 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) "," (Num 1) ")" )))) ; theorem :: PDIFF_5:29 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) "," (Num 2) ")" )))) ; theorem :: PDIFF_5:30 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u")) "," (Num 3) ")" )))) ; theorem :: PDIFF_5:31 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) "," (Num 1) ")" )))) ; theorem :: PDIFF_5:32 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) "," (Num 2) ")" )))) ; theorem :: PDIFF_5:33 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u")) "," (Num 3) ")" )))) ; theorem :: PDIFF_5:34 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k7_pdiff_5 :::"hpartdiff31"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) "," (Num 1) ")" )))) ; theorem :: PDIFF_5:35 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k8_pdiff_5 :::"hpartdiff32"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) "," (Num 2) ")" )))) ; theorem :: PDIFF_5:36 (Bool "for" (Set (Var "u")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u")))) "holds" (Bool (Set ($#k9_pdiff_5 :::"hpartdiff33"::: ) "(" (Set (Var "f")) "," (Set (Var "u")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u")) "," (Num 3) ")" )))) ; theorem :: PDIFF_5:37 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:38 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:39 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:40 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:41 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:42 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:43 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k7_pdiff_5 :::"hpartdiff31"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:44 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k8_pdiff_5 :::"hpartdiff32"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:45 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "N")) "being" ($#m1_rcomp_1 :::"Neighbourhood"::: ) "of" (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0"))) "st" (Bool (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "N")) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" )))) "holds" (Bool "for" (Set (Var "h")) "being" ($#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 "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")) ")" ) ($#k1_tarski :::"}"::: ) )) & (Bool (Set ($#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 "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" )) "is" ($#v2_comseq_2 :::"convergent"::: ) ) & (Bool (Set ($#k9_pdiff_5 :::"hpartdiff33"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_seq_2 :::"lim"::: ) (Set "(" (Set "(" (Set (Var "h")) ($#k37_valued_1 :::"""::: ) ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set "(" (Set (Var "h")) ($#k3_valued_1 :::"+"::: ) (Set (Var "c")) ")" ) ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" (Set "(" ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ")" ) ($#k8_funct_2 :::"/*"::: ) (Set (Var "c")) ")" ) ")" ) ")" ))) ")" )))))) ; theorem :: PDIFF_5:46 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:47 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:48 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:49 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:50 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:51 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k3_valued_1 :::"+"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:52 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:53 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:54 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:55 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:56 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:57 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k47_valued_1 :::"-"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f1")) "," (Set (Var "u0")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f2")) "," (Set (Var "u0")) ")" ")" ))) ")" ))) ; theorem :: PDIFF_5:58 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k1_pdiff_5 :::"hpartdiff11"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:59 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_pdiff_5 :::"hpartdiff12"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:60 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k3_pdiff_5 :::"hpartdiff13"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:61 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k4_pdiff_5 :::"hpartdiff21"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:62 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k5_pdiff_5 :::"hpartdiff22"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:63 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k6_pdiff_5 :::"hpartdiff23"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:64 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 1) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k7_pdiff_5 :::"hpartdiff31"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:65 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 2) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k8_pdiff_5 :::"hpartdiff32"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:66 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u0")))) "holds" (Bool "(" (Bool (Set (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)) & (Bool (Set ($#k11_pdiff_1 :::"partdiff"::: ) "(" (Set "(" (Set (Var "r")) ($#k26_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) ")" ) "," (Set (Var "u0")) "," (Num 3) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k9_pdiff_5 :::"hpartdiff33"::: ) "(" (Set (Var "f")) "," (Set (Var "u0")) ")" ")" ))) ")" )))) ; theorem :: PDIFF_5:67 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)))) ; theorem :: PDIFF_5:68 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)))) ; theorem :: PDIFF_5:69 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 1) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 1) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)))) ; theorem :: PDIFF_5:70 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)))) ; theorem :: PDIFF_5:71 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)))) ; theorem :: PDIFF_5:72 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 2) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 2) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)))) ; theorem :: PDIFF_5:73 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 3) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 3) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 1)))) ; theorem :: PDIFF_5:74 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 3) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 3) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 2)))) ; theorem :: PDIFF_5:75 (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Var "f1")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u0"))) & (Bool (Set (Var "f2")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f1")) "," (Num 3) ")" ")" ) ($#k20_valued_1 :::"(#)"::: ) (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f2")) "," (Num 3) ")" ")" )) ($#r3_pdiff_1 :::"is_partial_differentiable_in"::: ) (Set (Var "u0")) "," (Num 3)))) ; theorem :: PDIFF_5:76 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r1_pdiff_5 :::"is_hpartial_differentiable`11_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:77 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r2_pdiff_5 :::"is_hpartial_differentiable`12_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:78 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r3_pdiff_5 :::"is_hpartial_differentiable`13_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 1) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:79 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r4_pdiff_5 :::"is_hpartial_differentiable`21_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:80 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r5_pdiff_5 :::"is_hpartial_differentiable`22_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:81 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r6_pdiff_5 :::"is_hpartial_differentiable`23_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 2) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:82 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r7_pdiff_5 :::"is_hpartial_differentiable`31_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 1) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 1) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:83 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r8_pdiff_5 :::"is_hpartial_differentiable`32_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 2) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 2) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ; theorem :: PDIFF_5:84 (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set "(" ($#k1_euclid :::"REAL"::: ) (Num 3) ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) (Bool "for" (Set (Var "u0")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Num 3)) "st" (Bool (Bool (Set (Var "f")) ($#r9_pdiff_5 :::"is_hpartial_differentiable`33_in"::: ) (Set (Var "u0")))) "holds" (Bool (Set ($#k1_pdiff_2 :::"SVF1"::: ) "(" (Num 3) "," (Set "(" ($#k1_pdiff_3 :::"pdiff1"::: ) "(" (Set (Var "f")) "," (Num 3) ")" ")" ) "," (Set (Var "u0")) ")" ) ($#r1_fcont_1 :::"is_continuous_in"::: ) (Set (Set "(" ($#k1_pdiff_1 :::"proj"::: ) "(" (Num 3) "," (Num 3) ")" ")" ) ($#k1_seq_1 :::"."::: ) (Set (Var "u0")))))) ;