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<div class="add">

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">ADD</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:50"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">f1</font>, <font color="Olive">f2</font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font> holds  <font color="Maroon">ADD</font> <a href="binop_1.html#K1">.</a> <font color="Olive">f1</font>,<font color="Olive">f2</font> <a href="hidden.html#R1">=</a> <font color="Olive">f1</font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Olive">f2</font>
 ;<br/>

<b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E2:50"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">ADDF</font> = <font color="Maroon">ADD</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">F</font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<b>set </b><font color="Maroon">Fu</font> =  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:50_1_1_1"/><b>then </b>
( <font color="Maroon">ADD</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, </i>;<br/><a NAME="E3:50_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
( <font color="Maroon">ADDF</font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">ADD</font> &amp;  ex <font color="Olive">x</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">x</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">ADD</font> )
 <b>by </b><i><a class="ref" href="finsop_1.html#D1">FINSOP_1:def 1</a>, <a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/><a NAME="E4:50_1_1_1"/><b>then </b>
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">ADD</font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">ADD</font>
 <b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/><b>hence </b><a NAME="E5:50_1_1_1"/>
<font color="Maroon">ADD</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">F</font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:50_1_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/>

<a NAME="E2:50_1_1_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 0
 
;<br/>
<a NAME="E3:50_1_1_2_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E4:50_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nagata_1.html#E1"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:50_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 1 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ADD</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> ) &amp; <font color="Maroon">ADD</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finsop_1.html#T2">FINSOP_1:2</a></i>;<br/><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>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font> );<br/><a NAME="E5:50_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 <b>by </b><i>, </i>;<br/><a NAME="E6:50_1_1_2"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2_2"><b>proof </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="E1:50_1_1_2_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:50_1_1_2_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E3:50_1_1_2_2"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E4:50_1_1_2_2"/><b>then </b><i><font color="Green">E79</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:50_1_1_2_2"/>
1 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:50_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:50_1_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E8:50_1_1_2_2"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">fn</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>, <font color="Maroon">Fn1</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E10:50_1_1_2_2"/>
( <font color="Maroon">Fn1</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; <font color="Maroon">fn</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">ADD</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i>, , <a class="txt" href="nagata_1.html#E2"><i><font color="Green">Lemma34</font></i></a>, , </i>;<br/>
<a NAME="E11:50_1_1_2_2"/><b>then </b>
( <font color="Maroon">fn</font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">Fn1</font> is <a href="pscomp_1.html#V9">continuous</a> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">fn</font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">Fn1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:50_1_1_2_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:50_1_1_2"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S9">NAT_1:sch 9</a>(, <a class="txt" href="nagata_1.html#E1:50"><i><font color="Green">E33</font></i></a>);<br/></i><b>hence </b><a NAME="E8:50_1_1_2"/>
<font color="Maroon">ADD</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">F</font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">ADD</font> <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">F</font> is  <a href="pscomp_1.html#V9">continuous</a> <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>
 ;<br/>


</div>
