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<div class="add">

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="midsp_1.html#NM1">MidSp</a>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = 1 as    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <b>by </b><i><a class="ref" href="midsp_3.html#T24" target="_self">Th24</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">a<sub>1</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E4:43_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="midsp_3.html#G1" target="_self">ReperAlgebraStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="midsp_1.html#U1">MIDPOINT</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> #) as  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 <b>by </b><i><a class="txt" href="midsp_3.html#E8"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<a NAME="E7:43_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>9</sub></font><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <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>9</sub></font> st ( for <font color="Olive">b<sub>5</sub></font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Olive">b<sub>1</sub></font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>5</sub></font></span>)</span> ) holds <br/><font color="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></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="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>9</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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>9</sub></font>;<br/>



<b>assume </b><a NAME="E1:43_1_1_1"/>
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 <font color="Maroon">c<sub>10</sub></font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 ;<br/>

<i><font color="Green">E27</font></i>:  <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="midsp_3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="midsp_3.html#K2" target="_self">*&gt;</a></span> <a href="midsp_3.html#K3" target="_self">^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E4:43_1_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/>
 <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="midsp_3.html#K2" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="midsp_3.html#K2" target="_self">*&gt;</a></span> <a href="midsp_3.html#K3" target="_self">^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E4:43_1_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T58">FINSEQ_1:58</a></i>
;<br/>
<b>hence </b><a NAME="E4:43_1_1_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_3.html#E2:43_1_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E8:43_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="midsp_3.html#NR1" target="_self">is_invariance</a> 
 <b>by </b><i><a class="txt" href="midsp_3.html#E7:43_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="midsp_3.html#D5" target="_self">Def5</a></i>;<br/>


</div>
