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<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span><br/>  for <font color="Olive" title="b2">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b2">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">X</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Olive" title="b1">f</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Olive" title="b2">X</font>,<font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b1">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Olive" title="b2">X</font>,<font color="Olive" title="b4">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b2">X</font> ) ) holds <br/>( <font color="Olive" title="b1">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b3">x</font> &amp; (  for <font color="Olive" title="b4">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b5">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b5">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b6">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b6">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b4">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b5">w</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span><br/>  for <font color="Olive" title="b2">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">i</font> &amp; <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Olive" title="b1">X</font>,<font color="Olive" title="b3">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b1">X</font> ) ) holds <br/>( <font color="Maroon" title="c2">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b2">x</font> &amp; (  for <font color="Olive" title="b3">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b4">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b4">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b5">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b5">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b4">w</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Maroon" title="c3">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b2">i</font> &amp; <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Olive" title="b2">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c3">X</font> ) ) holds <br/>( <font color="Maroon" title="c2">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b1">x</font> &amp; (  for <font color="Olive" title="b2">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b3">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b3">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b4">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b4">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b3">w</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c3">X</font> ) ) implies ( <font color="Maroon" title="c2">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font> &amp; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b2">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">w</font> ) ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:68"/><span class="lab"><font color="Green" title="E62">A1</font></span>: 
( <font color="Maroon" title="c3">X</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> holds <br/>( <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Olive" title="b1">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c3">X</font> ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font> &amp; (  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b2">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">w</font> ) ) )</span><br/>

<a NAME="E2:68"/><span class="lab"><font color="Green" title="E63">A2</font></span>: 
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">One0</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T98" title="FINSEQ_2:th.98">FINSEQ_2:98</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">One</font> = <font color="Maroon" title="c5">One0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">f0</font> = <font color="Maroon" title="c2">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E63">A2</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">X0</font> = <font color="Maroon" title="c3">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x0</font> = <font color="Maroon" title="c4">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E8:68"/><span class="lab"><font color="Green" title="E64">A3</font></span>: 
<font color="Maroon" title="c8">X0</font> is  <a href="pdiff_7.html#V1" title="PDIFF_7:attr.1">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#D3" target="_self" title="PDIFF_7:def.3">Def3</a>, <a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E65">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> implies ( <font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>) <a href="pdiff_7.html#R4" title="PDIFF_7:pred.4">is_continuous_on</a> <font color="Maroon" title="c8">X0</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:68_1"/><span class="lab"><font color="Green" title="E66">A5</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>) <a href="pdiff_7.html#R4" title="PDIFF_7:pred.4">is_continuous_on</a> <font color="Maroon" title="c8">X0</font> )</span><br/><a NAME="E2:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A6</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#R7" title="PDIFF_1:pred.7">is_partial_differentiable_on</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:68_1"/><span class="lab"><font color="Green" title="E68">A7</font></span>: 
<font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#T33" target="_self" title="PDIFF_7:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>) <a href="pdiff_7.html#R4" title="PDIFF_7:pred.4">is_continuous_on</a> <font color="Maroon" title="c8">X0</font></span><br/><a NAME="E4:68_1"/><span class="lab"><font color="Green" title="E69">A8</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c10">i</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a></span>;<br/><a NAME="E5:68_1"/><span class="lab"><font color="Green" title="E70">A9</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X0</font> &amp; (  for <font color="Olive" title="b1">x0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K12" title="PDIFF_1:func.12">partdiff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Olive" title="b1">x0</font>,<font color="Maroon" title="c10">i</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#D5" target="_self" title="PDIFF_7:def.5">Def5</a>, <a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a></span>;<br/><a NAME="E6:68_1"/><span class="lab"><font color="Green" title="E71">A10</font></span>: 
 for <font color="Olive" title="b1">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">z</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:68_1_1"/><span class="lab"><font color="Green" title="E72">A11</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font> )</span><br/>

<a NAME="E2:68_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">f0</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c10">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E66">A5</font></span></a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="pdiff_7.html#T34" target="_self" title="PDIFF_7:th.34">Th34</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">g</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>, <font color="Maroon" title="c13">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:68_1_1"/><span class="lab"><font color="Green" title="E73">A12</font></span>: 
( <font color="Maroon" title="c7">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g</font> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> &amp;  <a href="pdiff_1.html#K12" title="PDIFF_1:func.12">partdiff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c10">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c12">g</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D14" title="PDIFF_1:def.14">PDIFF_1:def 14</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c13">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:68_1_1"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font></span><br/>

<span class="kw">thus </span><a NAME="E6:68_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="txt" href="#E1:68_1_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="pdiff_7.html#D5" target="_self" title="PDIFF_7:def.5">Def5</a>, <a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E68">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E7:68_1"/>
 for <font color="Olive" title="b1">z0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><br/>  for <font color="Olive" title="b2">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> holds <br/> ex <font color="Olive" title="b3">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> &amp; (  for <font color="Olive" title="b4">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b4">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Olive" title="b1">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">s</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">r</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">z0</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c11">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> ex <font color="Olive" title="b2">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> &amp; (  for <font color="Olive" title="b3">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b3">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">s</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c12">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> implies  ex <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; (  for <font color="Olive" title="b2">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b2">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:68_1_2"/><span class="lab"><font color="Green" title="E72">A13</font></span>: 
( <font color="Maroon" title="c11">z0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X0</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> &amp; (  for <font color="Olive" title="b2">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b2">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">y0</font> = <font color="Maroon" title="c11">z0</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E4:68_1_2"/><span class="lab"><font color="Green" title="E73">A14</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> &amp; (  for <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  st <font color="Olive" title="b1">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c13">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> holds <br/><span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1"><span class="lab"><font color="Green" title="E69">A8</font></span></a>, <a class="txt" href="#E1:68_1_2"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="nfcont_1.html#T19" title="NFCONT_1:th.19">NFCONT_1:19</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> &amp; (  for <font color="Olive" title="b1">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:68_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1_2"><span class="lab"><font color="Green" title="E73">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font></span><br/>

<span class="kw">thus </span><a NAME="E6:68_1_2"/>
 for <font color="Olive" title="b1">z1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  st <font color="Olive" title="b1">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> holds <br/><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">z1</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> implies <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:68_1_2_1"/><span class="lab"><font color="Green" title="E74">A15</font></span>: 
( <font color="Maroon" title="c15">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c15">z1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E3:68_1_2_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">z1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c11">z0</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">y1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c13">y0</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a>, <a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>;<br/>
<a NAME="E4:68_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A16</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">y1</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">y0</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_1_2_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="txt" href="#E4:68_1_2"><span class="lab"><font color="Green" title="E73">A14</font></span></a></span>;<br/>
<a NAME="E5:68_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="pdiff_1.html#K15" title="PDIFF_1:func.15">`partial|</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c10">i</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:68_1_2_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a>, <a class="ref" href="pdiff_1.html#D20" title="PDIFF_1:def.20">PDIFF_1:def 20</a></span>;<br/>
<a NAME="E6:68_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A17</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:68_1_2_1"><span class="lab"><font color="Green" title="E75">A16</font></span></a>, <a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E67">A6</font></span></a>, <a class="txt" href="#E1:68_1_2"><span class="lab"><font color="Green" title="E72">A13</font></span></a>, <a class="ref" href="pdiff_1.html#D20" title="PDIFF_1:def.20">PDIFF_1:def 20</a></span>;<br/>
<a NAME="E7:68_1_2_1"/><span class="lab"><font color="Green" title="E77">A18</font></span>: 
 ex <font color="Olive" title="b1">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c15">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">y1</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="txt" href="#E1:68_1_2_1"><span class="lab"><font color="Green" title="E74">A15</font></span></a></span>;<br/>
<a NAME="E8:68_1_2_1"/><span class="lab"><font color="Green" title="E78">A19</font></span>: 
 ex <font color="Olive" title="b1">y0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c11">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">y0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="txt" href="#E1:68_1_2"><span class="lab"><font color="Green" title="E72">A13</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">PDP</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" title="LOPBAN_1:def.9">LOPBAN_1:def 9</a></span>;<br/>
<a NAME="E10:68_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">PDP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">One</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T40" title="LOPBAN_1:th.40">LOPBAN_1:40</a></span>;<br/>
<a NAME="E11:68_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A20</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">One</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T32" title="LOPBAN_1:th.32">LOPBAN_1:32</a></span>;<br/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">One</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c5">One0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a></span>
<br/>.= 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="toprealc.html#T18" title="TOPREALC:th.18">TOPREALC:18</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<a NAME="E13:68_1_2_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c16">y1</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c13">y0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c6">One</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:68_1_2_1"><span class="lab"><font color="Green" title="E79">A20</font></span></a>, <a class="txt" href="#E6:68_1_2_1"><span class="lab"><font color="Green" title="E76">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:68_1_2_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">z1</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c11">z0</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:68_1_2_1"><span class="lab"><font color="Green" title="E77">A18</font></span></a>, <a class="txt" href="#E8:68_1_2_1"><span class="lab"><font color="Green" title="E78">A19</font></span></a>, <a class="ref" href="real_ns1.html#T1" title="REAL_NS1:th.1">REAL_NS1:1</a>, <a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:68_1"/>
<font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#K2" title="PDIFF_7:func.2">`partial|</a> (<font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c10">i</font>) <a href="pdiff_7.html#R4" title="PDIFF_7:pred.4">is_continuous_on</a> <font color="Maroon" title="c8">X0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="pdiff_7.html#T38" target="_self" title="PDIFF_7:th.38">Th38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:68"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A21</font></span>: 
( <font color="Maroon" title="c7">f0</font> <a href="pdiff_1.html#R1" title="PDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c9">x0</font> &amp; (  for <font color="Olive" title="b1">h0</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font>  ex <font color="Olive" title="b2">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">h0</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K12" title="PDIFF_1:func.12">partdiff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c9">x0</font>,<font color="Olive" title="b3">i</font>)</span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c9">x0</font>)</span>)</span> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <font color="Olive" title="b1">h0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">w</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#T47" target="_self" title="PDIFF_7:th.47">Th47</a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E62">A1</font></span></a></span>;<br/>
<a NAME="E11:68"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> ex <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c7">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c9">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b1">g</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D7" title="PDIFF_1:def.7">PDIFF_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:68"/>
<font color="Maroon" title="c2">f</font> <a href="ndiff_1.html#R1" title="NDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b2">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">w</font> )</span><br/>

<a NAME="E13:68"/><span class="lab"><font color="Green" title="E67">A22</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> ex <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c7">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c9">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp;  <a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c9">x0</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">y</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:68"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="ref" href="pdiff_1.html#D8" title="PDIFF_1:def.8">PDIFF_1:def 8</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">h</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b1">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b2">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b1">w</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">h0</font> = <font color="Maroon" title="c10">h</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1<span class="kw"> such that </span><br/><a NAME="E3:68_2"/><span class="lab"><font color="Green" title="E68">A23</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">h0</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K12" title="PDIFF_1:func.12">partdiff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c9">x0</font>,<font color="Olive" title="b1">i</font>)</span>)</span> ) &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K8" title="PDIFF_1:func.8">diff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c9">x0</font>)</span>)</span> <a href="pdiff_6.html#K1" title="PDIFF_6:func.1">.</a> <font color="Maroon" title="c11">h0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c12">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#T47" target="_self" title="PDIFF_7:th.47">Th47</a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E65">A4</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c13">w</font> = <font color="Maroon" title="c12">w</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c13">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c13">w</font> )</span><br/><span class="kw">thus </span><a NAME="E4:68_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_2"><span class="lab"><font color="Green" title="E68">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c13">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c13">w</font> )</span><br/><span class="kw">thus </span><a NAME="E5:68_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Maroon" title="c13">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c13">w</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="68_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> implies <font color="Maroon" title="c13">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c14">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:68_2_1"/><span class="lab"><font color="Green" title="E69">A24</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c14">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/>

<a NAME="E2:68_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A25</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:68_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">f0</font> <a href="pdiff_7.html#R2" title="PDIFF_7:pred.2">is_partial_differentiable_on</a> <font color="Maroon" title="c8">X0</font>,<font color="Maroon" title="c14">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:68"><span class="lab"><font color="Green" title="E65">A4</font></span></a></span>;<br/>
<a NAME="E4:68_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">f0</font> <a href="pdiff_1.html#R4" title="PDIFF_1:pred.4">is_partial_differentiable_in</a> <font color="Maroon" title="c9">x0</font>,<font color="Maroon" title="c14">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="txt" href="#E8:68"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="txt" href="#E2:68_2_1"><span class="lab"><font color="Green" title="E70">A25</font></span></a>, <a class="ref" href="pdiff_7.html#T34" target="_self" title="PDIFF_7:th.34">Th34</a></span>;<br/>
<a NAME="E5:68_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A26</font></span>: 
 ex <font color="Olive" title="b1">g</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> ex <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> st <br/>( <font color="Maroon" title="c7">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; <font color="Maroon" title="c9">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> &amp;  <a href="pdiff_1.html#K12" title="PDIFF_1:func.12">partdiff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c9">x0</font>,<font color="Maroon" title="c14">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">y</font>,<font color="Maroon" title="c14">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D14" title="PDIFF_1:def.14">PDIFF_1:def 14</a></span>;<br/>
<span class="lab"><font color="Green" title="E72">A27</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">One</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">h0</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c5">One0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T3" title="REAL_NS1:th.3">REAL_NS1:3</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">h0</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> 1</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T47" title="RVSUM_1:th.47">RVSUM_1:47</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>

;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">PDP</font> =  <a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c14">i</font>) as   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">bounded</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#D9" title="LOPBAN_1:def.9">LOPBAN_1:def 9</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">h0</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K12" title="PDIFF_1:func.12">partdiff</a> (<font color="Maroon" title="c7">f0</font>,<font color="Maroon" title="c9">x0</font>,<font color="Maroon" title="c14">i</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">h0</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">PDP</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">One</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_2_1"><span class="lab"><font color="Green" title="E71">A26</font></span></a>, <a class="ref" href="real_ns1.html#T3" title="REAL_NS1:th.3">REAL_NS1:3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c14">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_2_1"><span class="lab"><font color="Green" title="E72">A27</font></span></a>, <a class="ref" href="lopban_1.html#D5" title="LOPBAN_1:def.5">LOPBAN_1:def 5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E9:68_2_1"/>
<font color="Maroon" title="c13">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c14">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68_2_1"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E3:68_2"><span class="lab"><font color="Green" title="E68">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E6:68_2"/>
<span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Maroon" title="c13">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_2"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="txt" href="#E13:68"><span class="lab"><font color="Green" title="E67">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:68"/>
 for <font color="Olive" title="b1">h</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>  ex <font color="Olive" title="b2">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 st <br/>(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> &amp; (  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">m</font> holds <br/><font color="Olive" title="b2">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K9" title="PDIFF_1:func.9">partdiff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">h</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) &amp; <span class="p1">(<span class="default"><a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">x</font>)</span>)</span> <a href="lopban_1.html#K17" title="LOPBAN_1:func.17">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_7.html#K5" title="EUCLID_7:func.5">Sum</a> <font color="Olive" title="b2">w</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
