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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E56">Th66</font></span>: <a NAME="T65"><span class="comment"><font color="firebrick">:: NDIFF13:65</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">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>  ex <font color="Olive" title="b3">I</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Olive" title="b2">G</font>,<font color="Olive" title="b1">F</font>)</span>)</span> st <br/>(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>) &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b2">G</font>,<font color="Olive" title="b1">F</font>) &amp; <font color="Olive" title="b3">I</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b3">I</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>) &amp;  ex <font color="Olive" title="b4">J</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Olive" title="b2">G</font>,<font color="Olive" title="b1">F</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">I</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive" title="b4">J</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b2">G</font>,<font color="Olive" title="b1">F</font>) &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b4">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>) &amp; <font color="Olive" title="b4">J</font> <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b2">G</font>,<font color="Olive" title="b1">F</font>) ) &amp; (  for <font color="Olive" title="b4">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16" title="LOPBAN_1:func.16">R_NormSpace_of_BoundedLinearOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>)</span>)</span>  st <font color="Olive" title="b4">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>) holds <br/><font color="Olive" title="b3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lopban13.html#K1" title="LOPBAN13:func.1">Inv</a> <font color="Olive" title="b4">u</font> ) &amp; (  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b3">I</font> <a href="ndiff_6.html#R1" title="NDIFF_6:pred.1">is_differentiable_on</a> <font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1, <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>) &amp;  <a href="ndiff_6.html#K6" title="NDIFF_6:func.6">diff</a> (<font color="Olive" title="b3">I</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>)</span>)</span>) <a href="nfcont_1.html#R3" title="NFCONT_1:pred.3">is_continuous_on</a>  <a href="lopban13.html#K2" title="LOPBAN13:func.2">InvertibleOperators</a> (<font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>) ) ) )</div></div>
