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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">x</font>, <font color="Olive">y</font> being <a href="facirc_1.html#V3">nonpair-yielding</a>  <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">a<sub>1</sub></font> holds   <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">y</font></span>)</span>;<br/>
<a NAME="E1:43"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be  <a href="facirc_1.html#V3">nonpair-yielding</a>  <a href="circcomb.html#M1">FinSeqLen</a> of 0, <font color="Maroon">y</font> be  <a href="facirc_1.html#V3">nonpair-yielding</a>  <a href="circcomb.html#M1">FinSeqLen</a> of 0;<br/>

<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> ;<br/>
<a NAME="E1:43_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="circcomb.html#D12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E2:43_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:43_1"/>
0 <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K7">1GateCircStr</a> <a href="facirc_2.html#K3" target="_self">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:43_1"/>
 <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default">0 <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="circcomb.html#T49">CIRCCOMB:49</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:43"/><i><font color="Green">E37</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>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</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="43_2"><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>assume </b><a NAME="E1:43_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">i</font>]
 ;<br/>

<b>let </b><font color="Maroon">x</font> be  <a href="facirc_1.html#V3">nonpair-yielding</a>  <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1, <font color="Maroon">y</font> be  <a href="facirc_1.html#V3">nonpair-yielding</a>  <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1;<br/>

<b>consider </b><font color="Maroon">x'</font> being  <a href="facirc_1.html#V3">nonpair-yielding</a>  <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">i</font>, <font color="Maroon">z1</font> being  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:43_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="facirc_1.html#K8">^</a> <span class="p1"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z1</font></span><a href="facirc_1.html#K5">*&gt;</a></span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y'</font> being  <a href="facirc_1.html#V3">nonpair-yielding</a>  <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">i</font>, <font color="Maroon">z2</font> being  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:43_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="facirc_1.html#K8">^</a> <span class="p1"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z2</font></span><a href="facirc_1.html#K5">*&gt;</a></span>
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">S</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>;<br/>
<a NAME="E6:43_2"/><i><font color="Green">E59</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:43_2"/><i><font color="Green">E60</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z1</font></span><a href="facirc_1.html#K5">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <span class="p1"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z1</font></span><a href="facirc_1.html#K5">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">z1</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E8:43_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E9:43_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</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#R1">=</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="facirc_2.html#D1" target="_self">Def1</a>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E10:43_2"/><i><font color="Green">E63</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z2</font></span><a href="facirc_1.html#K5">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp; <span class="p1"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z2</font></span><a href="facirc_1.html#K5">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">z2</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/>
<a NAME="E11:43_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="circcomb.html#D12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E12:43_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</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#R1">=</a> <font color="Maroon">z2</font>
 
<b>by </b><i>, , <a class="ref" href="facirc_2.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<a NAME="E13:43_2"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#T100">ENUMSET1:100</a></i>;<br/>
<i><font color="Green">E67</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z1</font></span><a href="facirc_1.html#K5">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="facirc_2.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z1</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/>
<i><font color="Green">E71</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2"><a href="facirc_1.html#K5">&lt;*</a><span class="default"><font color="Maroon">z2</font></span><a href="facirc_1.html#K5">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T44">FINSEQ_1:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z2</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>
;<br/>
<a NAME="E16:43_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">z1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z2</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="facirc_1.html#K3">'&amp;'</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">z1</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="facirc_1.html#K3">'&amp;'</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="facirc_1.html#K3">'&amp;'</a> </span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="facirc_1.html#K1">'xor'</a> </span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E17:43_2"/>
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font> <a href="finseq_1.html#K7">^</a> <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/>
<a NAME="E18:43_2"/><b>then </b>
<font color="Maroon">i</font> <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="facirc_2.html#D1" target="_self">Def1</a>, , </i>;<br/>
<b>hence </b> <a href="msafree2.html#K2">InputVertices</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="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font></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> <span class="p3">(<span class="default"><a href="facirc_1.html#K23">BitAdderWithOverflowStr</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></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">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, , <a class="txt" href="facirc_2.html#E1:43"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></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> <span class="p3">(<span class="default"><a href="facirc_1.html#K23">BitAdderWithOverflowStr</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font>,<span class="p4">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></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">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="enumset1.html#K1">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font>,<span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="enumset1.html#K1">}</a></span> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="facirc_2.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i>, <a class="ref" href="enumset1.html#T136">ENUMSET1:136</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z1</font></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z2</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z1</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z2</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x'</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z1</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z2</font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="facirc_2.html#E2:43"><i><font color="Green">E37</font></i></a>, <a class="txt" href="facirc_2.html#E1:43_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:43"/>
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>1</sub></font>[<font color="Olive">i</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
