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<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">z</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green">E18</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font> or  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 </span><b>by </b><i>, <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>suppose </b></a><a NAME="E1:19_1_1"/>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_1"/>
 ex <font color="Olive">y</font> being  <a href="supinf_1.html#NM3">R_eal</a> st <br/>(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Olive">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_2"><b>suppose </b></a><a NAME="E1:19_1_2"/><i><font color="Green">E19</font></i>: 
 <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">y</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:19_1_2"/><i><font color="Green">E22</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> )
 <b>by </b><i>, </i>;<br/><b>take </b>
<font color="Maroon">y</font>
;<br/><a NAME="E4:19_1_2"/>
<font color="Maroon">y</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:19_1_2"/>
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 <b>by </b><i>, , <a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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