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<div class="add">

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">m</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">RAS</font> be    <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c4">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c3">RAS</font>;<br/>







<b>thus </b><a NAME="E1:52"/>
( <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Maroon" title="c2">m</font> implies for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</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" title="E34">A1</font></i>: 
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Maroon" title="c2">m</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>;<br/>
<b>set </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<b>set </b><font color="Maroon" title="c8">p'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">a</font>;<br/>
<a NAME="E3:52_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T40" title="MIDSP_2:th.40">MIDSP_2:40</a></i>;<br/>
<a NAME="E4:52_1"/><b>then </b><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T29" target="_self" title="MIDSP_3:th.29">Th29</a></i>;<br/>
<a NAME="E5:52_1"/>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:52_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="midsp_3.html#E3:52_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="midsp_3.html#D6" target="_self" title="MIDSP_3:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E6:52_1"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E4:52_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="midsp_3.html#T27" target="_self" title="MIDSP_3:th.27">Th27</a></i>;<br/>


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

<b>thus </b><a NAME="E2:52"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font> ) implies <font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Maroon" title="c2">m</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" title="E34">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font>
 ;<br/>

<a NAME="E2:52_2"/>
for <font color="Olive" title="b1">a</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font><br/> for <font color="Olive" title="b2">p</font> being  <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font> holds   <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</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" title="c5">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>;<br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font>;<br/>



<b>set </b><font color="Maroon" title="c7">v</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font>;<br/>
<b>set </b><font color="Maroon" title="c8">x'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span>;<br/>
<a NAME="E1:52_2_1"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></i>;<br/>
<a NAME="E2:52_2_1"/><b>then </b><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T28" target="_self" title="MIDSP_3:th.28">Th28</a></i>;<br/>
<a NAME="E3:52_2_1"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">p</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><a href="midsp_2.html#K11" title="MIDSP_2:func.11">0.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:52_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="txt" href="midsp_3.html#E1:52_2_1"><i><font color="Green" title="E35">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_1"/>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E2:52_2_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="midsp_3.html#T26" target="_self" title="MIDSP_3:th.26">Th26</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:52_2"/>
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R1" title="MIDSP_3:pred.1">has_property_of_zero_in</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D6" target="_self" title="MIDSP_3:def.6">Def6</a></i>;<br/>


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


</div>
