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<b>let </b><font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:59"/><i><font color="Green">E27</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 or 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font> )
 </span><b>by </b><i><a class="txt" href="xreal_1.html#E7"><i><font color="Green">Lemma42</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_1"><b>suppose </b></a><a NAME="E1:59_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:59_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="xcmplx_0.html#K5">"</a>  <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>;<br/><a NAME="E3:59_1_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K5">"</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 ;<br/><a NAME="E4:59_1_1"/><b>then </b>
( <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>;<br/><b>hence </b><a NAME="E5:59_1_1"/>
<font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_1_2"><b>suppose </b></a><a NAME="E1:59_1_2"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:59_1_2"/>
<font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font>
 ;<br/><a NAME="E3:59_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E8"><i><font color="Green">Lemma43</font></i></a>, </i>;<br/><a NAME="E4:59_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="txt" href="xreal_1.html#E8"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/><b>hence </b><a NAME="E5:59_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="xreal_1.html#E7"><i><font color="Green">Lemma42</font></i></a>, <a class="txt" href="xreal_1.html#E8"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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