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<a NAME="E1:11_1_1"/><i><font color="Green" title="E6">A0</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  holds  <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/>

<a NAME="E1:11_1_1_1"/>
<a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">b</font>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#D2" target="_self" title="MESFUNC7:def.2">Def14</a></i>;<br/>
<b>hence </b><a NAME="E2:11_1_1_1"/>
<a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="ref" href="mesfunc7.html#D2" target="_self" title="MESFUNC7:def.2">Def14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:11_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a>  holds  <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span>)</span>,<font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ;<br/>

<a NAME="E1:11_1_1_2"/>
<a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#D2" target="_self" title="MESFUNC7:def.2">Def14</a></i>;<br/>
<a NAME="E2:11_1_1_2"/><b>then </b>
<a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#D2" target="_self" title="MESFUNC7:def.2">Def14</a></i>;<br/>
<a NAME="E3:11_1_1_2"/><b>then </b>
<a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="ref" href="extreal1.html#T23" title="EXTREAL1:th.23">EXTREAL1:23</a></i>;<br/>
<a NAME="E4:11_1_1_2"/><b>then </b>
<a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="extreal1.html#K2" title="EXTREAL1:func.2">*</a> <font color="Maroon" title="c2">b</font></span>)</span>,<font color="Maroon" title="c3">c</font>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#D2" target="_self" title="MESFUNC7:def.2">Def14</a></i>;<br/>
<b>hence </b><a NAME="E5:11_1_1_2"/>
<a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="ref" href="mesfunc7.html#D2" target="_self" title="MESFUNC7:def.2">Def14</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_1_1"/>
(  <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp;  <a href="mesfunc7.html#K3" title="MESFUNC7:func.3">multextreal</a>  is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> )
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:11_1_1"><i><font color="Green" title="E6">A0</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a>, <a class="ref" href="binop_1.html#D3" title="BINOP_1:def.3">BINOP_1:def 3</a></i>;<br/>


</div>
