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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E55">Th65</font></span>: <a NAME="T64"><span class="comment"><font color="firebrick">:: NDIFF13:64</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">E</font> being   <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a><br/>  for <font color="Olive" title="b2">F</font>, <font color="Olive" title="b3">G</font> being   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="lopban_1.html#NM2" title="LOPBAN_1:NM.2">RealBanachSpace</a><br/>  for <font color="Olive" title="b4">Z</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span><br/>  for <font color="Olive" title="b5">c</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b3">G</font><br/>  for <font color="Olive" title="b6">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">E</font><br/>  for <font color="Olive" title="b7">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">F</font>  st <font color="Olive" title="b4">Z</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <font color="Olive" title="b6">A</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <font color="Olive" title="b7">B</font> is  <a href="nfcont_1.html#V3" title="NFCONT_1:attr.3">open</a>  &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Olive" title="b6">A</font>,<font color="Olive" title="b7">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">Z</font> holds <br/> for <font color="Olive" title="b8">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b9">f</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>,<font color="Olive" title="b3">G</font><br/>  for <font color="Olive" title="b10">g</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">F</font>  st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Z</font> &amp; <font color="Olive" title="b9">f</font> <a href="ndiff_6.html#R1" title="NDIFF_6:pred.1">is_differentiable_on</a> <font color="Olive" title="b8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1,<font color="Olive" title="b4">Z</font> &amp;  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b9">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b4">Z</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b4">Z</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b10">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">A</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b10">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b7">B</font> &amp; <font color="Olive" title="b10">g</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b6">A</font> &amp; (  for <font color="Olive" title="b11">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">E</font>  st <font color="Olive" title="b11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">A</font> holds <br/><font color="Olive" title="b9">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b11">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">x</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">c</font> ) &amp; (  for <font color="Olive" title="b11">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">E</font><br/>  for <font color="Olive" title="b12">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>  st <font color="Olive" title="b11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">A</font> &amp; <font color="Olive" title="b12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b11">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> <a href="ndiff_7.html#K8" title="NDIFF_7:func.8">partdiff`2</a> (<font color="Olive" title="b9">f</font>,<font color="Olive" title="b12">z</font>) is  <a href="lopban13.html#V1" title="LOPBAN13:attr.1">invertible</a>  ) holds <br/>( <font color="Olive" title="b10">g</font> <a href="ndiff_6.html#R1" title="NDIFF_6:pred.1">is_differentiable_on</a> <font color="Olive" title="b8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1,<font color="Olive" title="b6">A</font> &amp;  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b10">g</font>,<span class="p1">(<span class="default"><font color="Olive" title="b8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b6">A</font>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a> <font color="Olive" title="b6">A</font> &amp; (  for <font color="Olive" title="b11">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">E</font><br/>  for <font color="Olive" title="b12">z</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K9" title="PRVECT_3:func.9">[:</a><span class="default"><font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">F</font></span><a href="prvect_3.html#K9" title="PRVECT_3:func.9">:]</a></span>  st <font color="Olive" title="b11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">A</font> &amp; <font color="Olive" title="b12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b11">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b10">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b11">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/> <a href="ndiff_1.html#K3" title="NDIFF_1:func.3">diff</a> (<font color="Olive" title="b10">g</font>,<font color="Olive" title="b11">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lopban13.html#K1" title="LOPBAN13:func.1">Inv</a> <span class="p3">(<span class="default"><a href="ndiff_7.html#K8" title="NDIFF_7:func.8">partdiff`2</a> (<font color="Olive" title="b9">f</font>,<font color="Olive" title="b12">z</font>)</span>)</span></span>)</span> <a href="ndiff_2.html#K2" title="NDIFF_2:func.2">*</a> <span class="p2">(<span class="default"><a href="ndiff_7.html#K7" title="NDIFF_7:func.7">partdiff`1</a> (<font color="Olive" title="b9">f</font>,<font color="Olive" title="b12">z</font>)</span>)</span></span>)</span> ) )</div></div>
