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<div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="supinf_1.html#NM3">R_eal</a>, <font color="Maroon">y</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="E1:7"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green">E19</font></i>: 
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:7"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font>
 ;<br/>

<b>then </b><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:7_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:7"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_2"/>
<font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font>
 ;<br/>

<b>then </b><font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font></span>)</span> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E3:7_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
( <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">z</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">z</font> &amp; <font color="Maroon">x</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y</font> <a href="supinf_2.html#K4">-</a> <font color="Maroon">z</font> )
 <b>by </b><i>, , <a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>


</div>
