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<b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><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">f</font> be    <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> ;<br/><b>let </b><font color="Maroon">g</font> be    <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> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font> )
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E41</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="E3:32"/><i><font color="Green">E42</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">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 0 ) &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">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 1 ) )
 ;<br/>

<a NAME="E4:32"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:32"/><b>then </b><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>let </b><font color="Maroon">h</font> be    <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> ;<br/>

<b>take </b><font color="Maroon">a</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>;<br/>

<b>take </b><font color="Maroon">b</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>;<br/><a NAME="E1:32_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><i><font color="Green">E74</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">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" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font>

;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">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" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</font>

;<br/><b>hence </b><a NAME="E4:32_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funcsdom.html#K8" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">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" target="_self">RealFuncExtMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K2" target="_self">.</a> <span class="p3"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:32"/>
<font color="Maroon">h</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">A</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">A</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="Maroon">a</font>,<font color="Maroon">f</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">A</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="Maroon">b</font>,<font color="Maroon">g</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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