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<b>reconsider </b><font color="Maroon">f</font> = the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">I</font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="mcart_1.html#K12">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="funct_2.html#T38">FUNCT_2:38</a></i>;<br/>
<a NAME="E2:35_1_1"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:35_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="mcart_1.html#K12">:]</a></span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:35_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="mcart_1.html#K12">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font>, <font color="Maroon">v</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font>

<br/>.= 
<font color="Maroon">u</font> <a href="group_1.html#K1">*</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:35_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:35_1_1"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">R</font> <a href="realset1.html#K1">||</a> <font color="Maroon">I</font> is   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/>


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