<?xml version="1.0"?>
<div class="add">

<a NAME="E1:19_1"/><i><font color="Green">E48</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0) is   <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:19_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="circcomb.html#R1">tolerates</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0)
 
<b>by </b><i><a class="ref" href="circcomb.html#T55">CIRCCOMB:55</a></i>;<br/>
<b>then </b><i><font color="Green">E76</font></i>:  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>1</sub></font>() <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0)</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>()</span>)</span></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> ex <font color="Olive">S1</font>, <font color="Olive">S2</font> being non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>) &amp; <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>1</sub></font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">S2</font> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S2</font> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">S2</font> is   <a href="relat_1.html#NM1">Relation</a> );<br/>
<a NAME="E4:19_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">S0</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>take </b><font color="Maroon">S1</font> = <font color="Maroon">F<sub>1</sub></font>() <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0);<br/>

<b>thus </b><a NAME="E1:19_1_2"/>
( <font color="Maroon">S0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(0) &amp; <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>) )
 <b>by </b><i>, </i>;<br/>

<a NAME="E2:19_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:19_1_2"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">B</font> =  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>() as  <a href="facirc_1.html#V2">without_pairs</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">R1</font> =  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">R2</font> =  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0) as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i>, </i>;<br/>
<a NAME="E6:19_1_2"/>
( <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0) &amp; <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0) <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:19_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">R2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:19_1_2"/>
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E9:19_1_2"/>
(  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S0</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span>,0)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> 0</span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> is   <a href="relat_1.html#NM1">Relation</a> )
 <b>by </b><i>, , <a class="ref" href="circcomb.html#T15">CIRCCOMB:15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19_1"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>given </b><font color="Maroon">S1</font> being  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> being  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <b> such that </b><a NAME="E1:19_1_3"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>) &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) )
 <b>and </b><br/><a NAME="E2:19_1_3"/>
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">i</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E3:19_1_3"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font>
 <b>and </b><br/><a NAME="E4:19_1_3"/><i><font color="Green">E83</font></i>: 
not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font> is <a href="facirc_1.html#V2">with_pair</a>
 <b>and </b><br/><a NAME="E5:19_1_3"/><i><font color="Green">E84</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font> is   <a href="relat_1.html#NM1">Relation</a>
 ;<br/>

<a NAME="E6:19_1_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S2</font> <a href="circcomb.html#R1">tolerates</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)
 
<b>by </b><i><a class="ref" href="circcomb.html#T55">CIRCCOMB:55</a></i>;<br/>
<a NAME="E7:19_1_3"/><i><font color="Green">E86</font></i>: 
 <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) is   <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i/>;<br/>
<a NAME="E8:19_1_3"/><i><font color="Green">E88</font></i>: 
not <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> is <a href="facirc_1.html#V2">with_pair</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:19_1_3"/><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>take </b>
<font color="Maroon">S2</font>
;<br/>

<b>take </b><font color="Maroon">S3</font> = <font color="Maroon">S2</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>);<br/>

<b>thus </b><a NAME="E10:19_1_3"/>
( <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) &amp; <font color="Maroon">S3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>) )
 <b>by </b><i>, </i>;<br/>

<b>thus </b><i><font color="Green">E90</font></i>:  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S3</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)</span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, , , </i>
;<br/>

<a NAME="E12:19_1_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) &amp; <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) )
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> =  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font>, <font color="Maroon">X2</font> = <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span> as  <a href="facirc_1.html#V2">without_pairs</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E14:19_1_3"/><i><font color="Green">E111</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X2</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> =  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font>, <font color="Maroon">X2</font> =  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i>, </i>;<br/>
<a NAME="E16:19_1_3"/>
 <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, <a class="ref" href="circcomb.html#T15">CIRCCOMB:15</a></i>;<br/>
<b>hence </b><a NAME="E17:19_1_3"/>
( <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S3</font> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S3</font> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S3</font> is   <a href="relat_1.html#NM1">Relation</a> )
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:19_1"/><i><font color="Green">E112</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<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>consider </b><font color="Maroon">S1</font> being  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> being  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <b> such that </b><br/><a NAME="E8:19_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>) &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>and </b><br/><a NAME="E9:19_1"/>
( <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S2</font> is   <a href="relat_1.html#NM1">Relation</a> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">S1</font>
;<br/>

<b>take </b>
<font color="Maroon">S2</font>
;<br/>

<b>thus </b><a NAME="E10:19_1"/>
( <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">n</font>) &amp; <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> )
 <b>by </b><i/>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_4"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or  ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_4_1"><b>suppose </b></a><a NAME="E1:19_1_4_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_4_1"/>
(  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font> is <a href="facirc_1.html#V2">with_pair</a> )
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_4_2"><b>suppose </b></a><a NAME="E1:19_1_4_2"/>
 ex <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:19_1_4_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:19_1_4_2"/>
 ex <font color="Olive">T1</font>, <font color="Olive">T2</font> being non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>) &amp; <font color="Olive">T2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">T2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Olive">T1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<font color="Maroon">i</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">T2</font> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">T2</font> is <a href="facirc_1.html#V2">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">T2</font> is   <a href="relat_1.html#NM1">Relation</a> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:19_1_4_2"/>
(  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">S1</font> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">S1</font> is <a href="facirc_1.html#V2">with_pair</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
