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<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">m</font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">n</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">RAS</font> <a href="midsp_3.html#R2" target="_self">is_semi_additive_in</a> <font color="Maroon">m</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>;<br/><b>let </b><font color="Maroon">d</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="E2:28"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<font color="Maroon">d</font>
 ;<br/>

<a NAME="E3:28"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">q</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">qq</font> = <font color="Maroon">q</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span>;<br/>
<a NAME="E4:28"/>
<font color="Maroon">q</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">pp</font> = <font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span>;<br/>
<a NAME="E1:28_1"/>
for <font color="Olive">k</font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">n</font> holds  <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">n</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_1_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">m</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font> = 
<font color="Maroon">q</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font>
<b>by </b><i>, , <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:28_1"/>
<font color="Maroon">q</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:28"/>
 <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> <font color="Maroon">m</font>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <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">a</font>,<font color="Maroon">q</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div>
