<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">RAS</font> be    <a href="midsp_3.html#M2" target="_self">ReperAlgebra</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="midsp_2.html#NM1">ATLAS</a> of <font color="Maroon">RAS</font>;<br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>thus </b><a NAME="E2:60"/>
( <font color="Maroon">RAS</font> <a href="midsp_3.html#R4" target="_self">is_alternative_in</a> <font color="Maroon">m</font> implies for <font color="Olive">x</font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">W</font> holds   <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive">x</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">RAS</font> <a href="midsp_3.html#R4" target="_self">is_alternative_in</a> <font color="Maroon">m</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">W</font>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">p'</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span>;<br/>
<a NAME="E3:60_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T40">MIDSP_2:40</a></i>;<br/>
<a NAME="E4:60_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E6"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="midsp_3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:60_1"/>
 <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E7"><i><font color="Green">Lemma41</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E6:60_1"/>
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:60"/>
( ( for <font color="Olive">x</font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">W</font> holds   <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive">x</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font> ) implies <font color="Maroon">RAS</font> <a href="midsp_3.html#R4" target="_self">is_alternative_in</a> <font color="Maroon">m</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_2"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">x</font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">W</font> holds   <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive">x</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font>
 ;<br/>

<a NAME="E2:60_2"/>
for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font><br/> for <font color="Olive">p</font> being  <a href="midsp_3.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">RAS</font><br/> for <font color="Olive">pm</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font>  st <font color="Olive">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">pm</font> holds <br/> <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Olive">a</font>,<span class="p1">(<span class="default"><font color="Olive">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">pm</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="midsp_3.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">RAS</font>;<br/><b>let </b><font color="Maroon">pm</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font>;<br/>





<b>assume </b><a NAME="E1:60_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pm</font>
 ;<br/>

<b>set </b><font color="Maroon">x</font> = <font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">v</font> = <font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">x'</font> = <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span>;<br/>
<a NAME="E2:60_2_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/>
<a NAME="E3:60_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E6"><i><font color="Green">Lemma40</font></i></a>, </i>;<br/>
<a NAME="E4:60_2_1"/>
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E7"><i><font color="Green">Lemma41</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E5:60_2_1"/>
 <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">pm</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="txt" href="midsp_3.html#E8"><i><font color="Green">Lemma44</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_2"/>
<font color="Maroon">RAS</font> <a href="midsp_3.html#R4" target="_self">is_alternative_in</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>


</div>
