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<span class="kw">let </span><font color="Maroon" title="c1">y</font>, <font color="Maroon" title="c2">x</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"> ( (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> ) iff <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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="c1">y</font> )</span><br/>



<a NAME="E1:9"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
(  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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="c1">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">and </span><br/><a NAME="E2:9_1"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</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"><font color="Maroon" title="c2">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="c1">y</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> or  not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</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"><font color="Maroon" title="c2">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="c1">y</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:9_1_1_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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="c1">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_1_1_2"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</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"><font color="Maroon" title="c2">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="c1">y</font></span><br/><div class="add"><a NAME="E2:9_1_1_2"/>
 <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="xxreal_3.html#T60" title="XXREAL_3:th.60">XXREAL_3:60</a></span>;<br/><span class="kw">hence </span><a NAME="E3:9_1_1_2"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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="c1">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="extreal1.html#D1" title="EXTREAL1:def.1">EXTREAL1:def 1</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:9"/>
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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="c1">y</font> implies (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:9_2"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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="c1">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:9_2_1"/>
( <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">y</font> or <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">y</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_2"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:9_2_1_1"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:9_2_1_1"/>
(  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal2.html#T9" target="_self" title="EXTREAL2:th.9">Th57</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:9_2_1_2"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:9_2_1_2"/>
(  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="extreal2.html#T10" target="_self" title="EXTREAL2:th.10">Th58</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:9"/>
( (  <a href="supinf_2.html#K2" title="SUPINF_2:func.2">-</a> <font color="Maroon" title="c1">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">y</font> ) iff <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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="c1">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></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/>


</div>
