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

<b>set </b><font color="Maroon" title="c1">Q1</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c2">Q2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span> ;<br/>
<a NAME="E1:130_1_1"/><i><font color="Green" title="E85">A1</font></i>: 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:130_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/>

<a NAME="E2:130_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1 )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:130_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="comput_1.html#T34" target="_self" title="COMPUT_1:th.34">Th34</a>, <a class="ref" href="comput_1.html#T37" target="_self" title="COMPUT_1:th.37">Th37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:130_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="130_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:130_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">n</font>, <font color="Maroon" title="c5">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:130_1_1_2"/><i><font color="Green" title="E86">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Maroon" title="c5">i</font>
 <b>and </b><br/><a NAME="E4:130_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> )
 ;<br/>
<b>thus </b><a NAME="E5:130_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="comput_1.html#E3:130_1_1_2"><i><font color="Green" title="E86">A2</font></i></a>, <a class="ref" href="comput_1.html#T39" target="_self" title="COMPUT_1:th.39">Th39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:130_1_1"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="comput_1.html#K5" title="COMPUT_1:func.5">const</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<span class="p2">(<span class="default">1 <a href="comput_1.html#K6" title="COMPUT_1:func.6">succ</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="comput_1.html#K7" title="COMPUT_1:func.7">proj</a> <font color="Olive" title="b2">i</font></span>)</span> where <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="comput_1.html#K4" title="COMPUT_1:func.4">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>
 <b>by </b><i><a class="txt" href="comput_1.html#E1:130_1_1"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


</div>
