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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    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="midsp_3.html#M2" target="_self">ReperAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</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>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="midsp_2.html#NM1">ATLAS</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="midsp_2.html#NM2">Vector</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</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>7</sub></font>;<br/>

















<b>assume </b><a NAME="E1:50"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E2:50"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:50_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_1"><b>suppose </b></a><a NAME="E1:50_1_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self">Def12</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:50_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="midsp_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:50"><i><font color="Green">E33</font></i></a>, <a class="txt" href="midsp_3.html#E1:50_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="midsp_3.html#T15" target="_self">Th15</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="50_1_1_1_2"><b>suppose </b></a><a NAME="E1:50_1_1_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self">Def12</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>4</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>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:50_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:50"><i><font color="Green">E33</font></i></a>, <a class="ref" href="midsp_3.html#D12" target="_self">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:50_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:50_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:50"/>
<font color="Maroon">c<sub>7</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T16" target="_self">Th16</a></i>;<br/>


</div>
