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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="midsp_3.html#M2" target="_self">ReperAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="midsp_2.html#NM1">ATLAS</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>thus </b><a NAME="E1:52"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R1" target="_self">has_property_of_zero_in</a> <font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font> holds  <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R1" target="_self">has_property_of_zero_in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font>;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E3:52_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T40">MIDSP_2:40</a></i>;<br/>
<a NAME="E4:52_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T29" target="_self">Th29</a></i>;<br/>
<a NAME="E5:52_1"/>
 <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:52_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="midsp_3.html#E3:52_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="midsp_3.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:52_1"/>
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E4:52_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="midsp_3.html#T27" target="_self">Th27</a></i>;<br/>


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

<b>thus </b><a NAME="E2:52"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font> holds  <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font> ) implies <font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R1" target="_self">has_property_of_zero_in</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:52_2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font> holds  <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:52_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="midsp_3.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> holds  <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="midsp_3.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>



<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<a NAME="E1:52_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/>
<a NAME="E2:52_2_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E3:52_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">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11">0.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:52_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="midsp_3.html#E1:52_2_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_1"/>
 <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E2:52_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="midsp_3.html#T26" target="_self">Th26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R1" target="_self">has_property_of_zero_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D6" target="_self">Def6</a></i>;<br/>


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


</div>
