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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:34"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:34"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>3</sub></font>,<a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:34"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) )
 <b>and </b><br/><a NAME="E5:34"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="ref" href="funcsdom.html#T28" target="_self">Th28</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E6:34"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>3</sub></font>,<a href="numbers.html#K1">REAL</a> ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="funcsdom.html#E1:34"><i><font color="Green">E27</font></i></a>, <a class="txt" href="funcsdom.html#E2:34"><i><font color="Green">E28</font></i></a>, <a class="txt" href="funcsdom.html#E4:34"><i><font color="Green">E29</font></i></a>, <a class="txt" href="funcsdom.html#E5:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funcsdom.html#T29" target="_self">Th29</a>, <a class="ref" href="funcsdom.html#T31" target="_self">Th31</a></i>;<br/>


</div>
