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<b>let </b><font color="Maroon">S</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">Z</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>





<b>assume </b><a NAME="E1:58"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">Z</font> is <a href="nfcont_1.html#V3">open</a>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:58"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:58"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="partfun2.html#K1">id</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E4:58"/><i><font color="Green">E47</font></i>: 
 <a href="lopban_1.html#K16">R_NormSpace_of_BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="normsp_1.html#G1">NORMSTR</a>(# <span class="p1">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K10">Zero_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K8">Add_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="rsspace.html#K9">Mult_</a> <span class="p2">(<span class="default"><a href="lopban_1.html#K10">BoundedLinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8">R_VectorSpace_of_LinearOperators</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="lopban_1.html#K15">BoundedLinearOperatorsNorm</a> <font color="Maroon">S</font>,<font color="Maroon">S</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lopban_1.html#D15">LOPBAN_1:def 15</a></i>;<br/>
<b>set </b><font color="Maroon">L</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>;<br/>
<a NAME="E5:58"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> is  <a href="lopban_1.html#V3">bounded</a> <a href="lopban_1.html#NM1">LinearOperator</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="lopban_2.html#T3">LOPBAN_2:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">L</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> as   <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">S</font></span>)</span> <b>by </b><i><a class="txt" href="ndiff_1.html#E3:58"><i><font color="Green">E46</font></i></a>, <a class="ref" href="lopban_1.html#D10">LOPBAN_1:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">R</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="funcop_1.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span> as   <a href="nfcont_1.html#NM1">PartFunc</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E8:58"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">r</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font> holds  <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E9:58"/><i><font color="Green">E50</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E10:58"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">R</font> is <a href="partfun1.html#V1">total</a>
 
<b>by </b><i><a class="ref" href="partfun1.html#D4">PARTFUN1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><i><font color="Green">E54</font></i>: <font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E8:58"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E4:58"><i><font color="Green">E47</font></i></a></i>
;<br/><a NAME="E2:58_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E8:58"><i><font color="Green">E48</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</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="Maroon">h</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="nfcont_1.html#T8">NFCONT_1:8</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E10:58"><i><font color="Green">E51</font></i></a>, <a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>
;<br/></div><b>end;</b></div><a NAME="E2:58_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="funct_1.html#V3">constant</a>
 <b>by </b><i><a class="ref" href="lopban_3.html#D5">LOPBAN_3:def 5</a></i>;<br/><b>hence </b><a NAME="E3:58_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a>
 <b>by </b><i/>;<br/><a NAME="E4:58_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E9:58"><i><font color="Green">E50</font></i></a></i>;<br/><b>hence </b><a NAME="E5:58_1"/>
 <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">h</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E10:58"><i><font color="Green">E51</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">S</font> <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:58_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><a NAME="E2:58_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Z</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E2:58"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E3:58_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E10:58"><i><font color="Green">E51</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:58_2"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="ndiff_1.html#E10:58"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>assume </b><a NAME="E1:58_3"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>then 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="E3:58_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">N</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:58"><i><font color="Green">E43</font></i></a>, </i>;<br/><a NAME="E4:58_3"/><i><font color="Green">E67</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>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:58_3"/>
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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E1:58_3_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 ;<br/>

<i><font color="Green">E69</font></i>: <font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> = 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E8:58"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E4:58"><i><font color="Green">E47</font></i></a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">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> = 
<font color="Maroon">x</font> <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>
<b>by </b><i><a class="txt" href="ndiff_1.html#E9:58"><i><font color="Green">E50</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E9:58"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<font color="Maroon">L</font> <a href="lopban_1.html#K17">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<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="Maroon">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="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ndiff_1.html#T2" target="_self">Th2</a>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:58_3"/>
<font color="Maroon">f</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:58"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> <a href="ndiff_1.html#R2" target="_self">is_differentiable_on</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:58"><i><font color="Green">E43</font></i></a>, , </i>;<br/>

<b>let </b><font color="Maroon">x0</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E16:58"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E17:58"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">f</font> <a href="ndiff_1.html#R1" target="_self">is_differentiable_in</a> <font color="Maroon">x0</font>
 
<b>by </b><i><a class="txt" href="ndiff_1.html#E10:58"><i><font color="Green">E51</font></i></a></i>;<br/>
<a NAME="E18:58"/><b>then </b>
 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">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">S</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">S</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="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/>;<br/>
<b>then 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="E20:58"/><i><font color="Green">E118</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>
 ;<br/>
<b>consider </b><font color="Maroon">N1</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E22:58"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">N1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="ndiff_1.html#E1:58"><i><font color="Green">E43</font></i></a>, , </i>;<br/>
<b>consider </b><font color="Maroon">N2</font> being    <a href="nfcont_1.html#M1">Neighbourhood</a> of <font color="Maroon">x0</font><b> such that </b><br/><a NAME="E24:58"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N1</font> &amp; <font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">N</font> )
 <b>by </b><i/>;<br/>
<a NAME="E25:58"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">N2</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E26:58"/><i><font color="Green">E122</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">N2</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>

<i><font color="Green">E123</font></i>: <font color="Maroon">R</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span> = 
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="ndiff_1.html#E8:58"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E4:58"><i><font color="Green">E47</font></i></a></i>
;<br/>
<b>assume </b><a NAME="E2:58_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N2</font>
 ;<br/>

<a NAME="E3:58_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">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> = 
<font color="Maroon">x</font> <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>
<b>by </b><i><a class="txt" href="ndiff_1.html#E9:58"><i><font color="Green">E50</font></i></a>, <a class="ref" href="ndiff_1.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font>
<b>by </b><i><a class="txt" href="ndiff_1.html#E9:58"><i><font color="Green">E50</font></i></a>, </i>
<br/>.= 
<font color="Maroon">L</font> <a href="lopban_1.html#K17">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<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="Maroon">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="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x0</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="ndiff_1.html#K4" target="_self">`|</a> <font color="Maroon">Z</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">x0</font> = 
 <a href="ndiff_1.html#K3" target="_self">diff</a> <font color="Maroon">f</font>,<font color="Maroon">x0</font>
<b>by </b><i>, , <a class="ref" href="ndiff_1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
<b>by </b><i>, , , </i>
;<br/>


</div>
