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<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">RAS</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="midsp_1.html#V2" title="MIDSP_1:attr.2">MidSp-like</a>  <a href="midsp_3.html#L1" title="MIDSP_3:struct.1">ReperAlgebraStr</a> of <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">RAS</font>;<br/><b>let </b><font color="Maroon" title="c4">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="c2">RAS</font>;<br/><b>let </b><font color="Maroon" title="c5">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c2">RAS</font>;<br/><b>let </b><font color="Maroon" title="c6">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="c5">W</font>;<br/>











<b>thus </b><a NAME="E1:37"/>
( <font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> implies <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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/><a NAME="E1:37_1_1"/>
<font color="Maroon" title="c5">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E1:37_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></i>;<br/><b>hence </b><a NAME="E2:37_1_1"/>
<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37_1"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:37"/>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">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/><a NAME="E1:37_2_1"/>
<font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c5">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E1:37_2"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</a></i>;<br/><b>hence </b><a NAME="E2:37_2_1"/>
<font color="Maroon" title="c5">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c3">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c7">m</font>
 <b>by </b><i><a class="ref" href="midsp_2.html#T39" title="MIDSP_2:th.39">MIDSP_2:39</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37_2"/>
<font color="Maroon" title="c5">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></i>;<br/>


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


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