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

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">x3</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/>







<b>assume </b><a NAME="E1:49"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K8">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="domain_1.html#K8">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> )
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> , <font color="Maroon">g</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> , <font color="Maroon">h</font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:49"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>( ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>and </b><br/><a NAME="E4:49"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>( ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> implies <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>and </b><br/><a NAME="E5:49"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>( ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x3</font> implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Olive">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x3</font> implies <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

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

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

<b>thus </b><a NAME="E6:49"/>
( ( for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a>  st <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="funcsdom.html#K9">RealFuncZero</a> <font color="Maroon">A</font> holds <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">c</font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( for <font color="Olive">h'</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">A</font>,<a href="numbers.html#K1">REAL</a>   ex <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">h'</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">c</font>,<font color="Maroon">h</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> ) )
 <b>by </b><i>, , , , , </i>;<br/>


</div>
