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<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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span>;<br/>



<a NAME="E1:44"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span> holds  <font color="Olive">x</font> <a href="group_1.html#K1">*</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> <a href="group_1.html#K1">*</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">IT</font> =  <a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font>;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font>, <font color="Maroon">g</font> = <font color="Maroon">y</font>, <font color="Maroon">h</font> = <font color="Maroon">z</font> as    <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>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> = 
<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> <font color="Maroon">f</font>,<font color="Maroon">g</font>

<br/>.= 
the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">x</font>

;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font> = 
<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#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font></span>)</span>,<font color="Maroon">h</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default">the <a href="rlvect_1.html#U1">add</a> of <span class="p2">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span>

;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span></span>)</span> = 
<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> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>

<br/>.= 
<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"><a href="funcsdom.html#K9" target="_self">RealFuncZero</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>

<b>thus </b><a NAME="E6:44"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span> st <font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">h</font> = <span class="p1">(<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="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <span class="p1">(<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="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">t</font>
;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">t</font> = 
<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> <font color="Maroon">f</font>,<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"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span></span>)</span>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:44"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:44"/>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="funcsdom.html#K10" target="_self">RealFuncUnit</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K10" target="_self">RealFuncUnit</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>

<b>hence </b><a NAME="E10:44"/>
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="funcsdom.html#K12" target="_self">RRing</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>

<b>thus </b><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<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> <font color="Maroon">g</font>,<font color="Maroon">h</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="funcsdom.html#K6" target="_self">RealFuncAdd</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funcsdom.html#K7" target="_self">RealFuncMult</a> <font color="Maroon">A</font></span>)</span> <a href="funcsdom.html#K1" target="_self">.</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>

;<br/>

<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">z</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/><br/>


</div>
