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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">w</font> be   <a href="supinf_1.html#NM1" title="SUPINF_1:NM.1">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">w</font> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c4">w</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:20"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> &amp; <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c3">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">w</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c4">w</font></span><br/>

<a NAME="E2:20"/><span class="kw">then </span>
(  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c2">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c4">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="extreal2.html#T12" target="_self" title="EXTREAL2:th.12">Th60</a></span>;<br/>
<a NAME="E3:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c3">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<a NAME="E4:20"/>
( <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="extreal2.html#T12" target="_self" title="EXTREAL2:th.12">Th60</a></span>;<br/>
<a NAME="E5:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c3">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c4">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T36" title="XXREAL_3:th.36">XXREAL_3:36</a></span>;<br/>
<a NAME="E6:20"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c2">z</font></span>)</span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c4">w</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_3.html#T25" title="XXREAL_3:th.25">XXREAL_3:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:20"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">z</font> <a href="supinf_2.html#K3" title="SUPINF_2:func.3">+</a> <font color="Maroon" title="c4">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E3:20"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="extreal2.html#T12" target="_self" title="EXTREAL2:th.12">Th60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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