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<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  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">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</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>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="midsp_2.html#NM1">ATLAS</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</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>5</sub></font>;<br/>











<b>thus </b><a NAME="E1:37"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> implies <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">E21</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:37_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E1:37_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="midsp_3.html#D12" target="_self">Def12</a></i>;<br/><b>hence </b><a NAME="E2:37_1_1"/>
<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <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="E3:37_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:37"/>
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></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">c<sub>7</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:37_2_1"/>
<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E1:37_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>;<br/><b>hence </b><a NAME="E2:37_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>
 <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="E3:37_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<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_3.html#D12" target="_self">Def12</a></i>;<br/>


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


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