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<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">RAS</font> be  non <a href="struct_0.html#V3">empty</a> <a href="midsp_1.html#V2">MidSp-like</a>  <a href="midsp_3.html#L1" target="_self">ReperAlgebraStr</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 2;<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/>





<div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">RAS</font> <a href="midsp_3.html#NR1" target="_self">is_invariance</a> 
 ;<br/><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">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</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>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">q</font> = <font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font>;<br/><i><font color="Green">E37</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">a</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span> = 
<font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span>
<b>by </b><i/>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_2"><b>now </b></a><div class="add"><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/><font color="Maroon">W</font> <a href="midsp_2.html#K9">.</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#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> = 
<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>,<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></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</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>, </i>
;<br/><b>hence </b><a NAME="E2:40_1_2"/>
<font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</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> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="midsp_1.html#K4">@</a> <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="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/></div><b>end;</b></div><a NAME="E4:40_1"/><b>then </b>
<font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font>,<font color="Maroon">x</font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<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></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:40_1"/>
 <a href="midsp_3.html#K8" target="_self">Phi</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="midsp_3.html#K8" target="_self">Phi</a> <font color="Maroon">b</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:40"/>
( ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font><br/> 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#K8" target="_self">Phi</a> <font color="Olive">a</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="midsp_3.html#K8" target="_self">Phi</a> <font color="Olive">b</font>,<font color="Olive">x</font> ) implies <font color="Maroon">RAS</font> <a href="midsp_3.html#NR1" target="_self">is_invariance</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font><br/> 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#K8" target="_self">Phi</a> <font color="Olive">a</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="midsp_3.html#K8" target="_self">Phi</a> <font color="Olive">b</font>,<font color="Olive">x</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="midsp_3.html#D5" target="_self">MIDSP_3:def 5</a><br/><b>let </b><font color="Maroon">b</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">q</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>assume </b><a NAME="E2:40_2"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">m</font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">n</font> holds  <font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">m</font></span>)</span>
 ;<br/>

<a NAME="E3:40_2"/>
<font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">b</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>proof </b></a><div class="add">

<div><i><font color="Green">E88</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_1"><b>now </b></a><div class="add"><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/><a NAME="E1:40_2_1_1"/>
<font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:40_2_1_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/><b>thus </b><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#K5" target="_self">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">q</font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<i><font color="Green">E90</font></i>: <font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font>,<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#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span></span>)</span> = 
 <a href="midsp_3.html#K8" target="_self">Phi</a> <font color="Maroon">a</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>

<br/>.= 
 <a href="midsp_3.html#K8" target="_self">Phi</a> <font color="Maroon">b</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>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font>,<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#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span></span>)</span>

;<br/>
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font>,<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#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">b</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font>,<span class="p3">(<span class="default"><font color="Maroon">W</font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">q</font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">W</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">W</font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">b</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">b</font>,<font color="Maroon">q</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E4:40_2_1"/>
<font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">b</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_2"/>
<font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">b</font>,<font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">a</font>,<font color="Maroon">p</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
( <font color="Maroon">RAS</font> <a href="midsp_3.html#NR1" target="_self">is_invariance</a>  iff for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">RAS</font><br/> 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#K8" target="_self">Phi</a> <font color="Olive">a</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="midsp_3.html#K8" target="_self">Phi</a> <font color="Olive">b</font>,<font color="Olive">x</font> )
 <b>by </b><i/>;<br/>


</div>
