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<b>let </b><font color="Maroon">X</font> be   <a href="lopban_1.html#NM2">RealBanachSpace</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">X</font>;<br/>



<b>given </b><font color="Maroon">n0</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:20"/><i><font color="Green">E50</font></i>: 
 <a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n0</font> is    <a href="nfcont_2.html#M1" target="_self">contraction</a> of <font color="Maroon">X</font>
 ;<br/>

<b>consider </b><font color="Maroon">xp</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:20"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n0</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xp</font>
 <b>and </b><br/><a NAME="E4:20"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n0</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/><font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 <b>by </b><i>, </i>;<br/>
<i><font color="Green">E53</font></i>: <span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n0</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n0</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xp</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n0</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font>
<b>by </b><i><a class="ref" href="funct_7.html#T71">FUNCT_7:71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n0</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xp</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/>
<a NAME="E6:20"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xp</font>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_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">X</font>;<br/><b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:20_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>;<br/>
<span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,0</span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T86">FUNCT_7:86</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
;<br/>
<a NAME="E2:20_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2_1_2"><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/><b>assume </b><a NAME="E1:20_2_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/><span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_7.html#T73">FUNCT_7:73</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/><b>hence </b><a NAME="E3:20_2_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:20_2_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:20_2_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Olive">n</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E3:20_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="abian.html#K1">iter</a> <font color="Maroon">f</font>,<font color="Maroon">n0</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E4:20_2"/>
<font color="Maroon">xp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:20"/>
 ex <font color="Olive">xp</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">xp</font> <a href="hidden.html#R1">=</a> <font color="Olive">xp</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> holds <br/><font color="Olive">xp</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 <b>by </b><i/>;<br/>


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