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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:132_1_1"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:132_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<a NAME="E2:132_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="comput_1.html#K6" target="_self">const</a> 0 or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="comput_1.html#K7" target="_self">succ</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:132_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T34" target="_self">Th34</a>, <a class="ref" href="comput_1.html#T37" target="_self">Th37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:132_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:132_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:132_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E4:132_1_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<b>thus </b><a NAME="E5:132_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:132_1_1_2"><i><font color="Green">E89</font></i></a>, <a class="ref" href="comput_1.html#T39" target="_self">Th39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:132_1_1"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default">0 <a href="comput_1.html#K6" target="_self">const</a> 0</span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K7" target="_self">succ</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="comput_1.html#K8" target="_self">proj</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a>, B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K5" target="_self">HFuncs</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:132_1_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


</div>
