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<b>consider </b><font color="Maroon">N</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E2:45_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">L</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span> ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>, <font color="Maroon">R</font> being   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:45_1_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E5:45_1_1"/>
 ex <font color="Olive">N</font> being   <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font> st <br/>( <font color="Olive">N</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp;  ex <font color="Olive">R</font> being  <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> st <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">N</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">x</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lopban_1.html#K17">.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Olive">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>


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