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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="supinf_1.html#NM3" title="SUPINF_1:NM.3">R_eal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> or <font color="Maroon" title="c2">y</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ) implies <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><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="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></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> <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#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> )</span><br/>



<b>assume </b><a NAME="E1:60"/><i><font color="Green" title="E35">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> or <font color="Maroon" title="c2">y</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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"><span class="p3"><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="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></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> <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#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span><br/>

<a NAME="E2:60"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
<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="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">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> <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#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T68" target="_self" title="EXTREAL2:th.68">Th68</a></i>;<br/>
<a NAME="E3:60"/><i><font color="Green" title="E37">A3</font></i>: 
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <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> <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="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="extreal2.html#T68" target="_self" title="EXTREAL2:th.68">Th68</a></i>;<br/>
<a NAME="E4:60"/>
<font color="Maroon" title="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T15" target="_self" title="EXTREAL2:th.15">Th15</a></i>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
<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="c2">y</font> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c1">x</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T66" target="_self" title="EXTREAL2:th.66">Th66</a></i>;<br/>
<a NAME="E6:60"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span> <a href="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><span class="p2"><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="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p2"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T15" target="_self" title="EXTREAL2:th.15">Th15</a></i>;<br/>
<a NAME="E7:60"/><b>then </b>
 <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <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#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">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>  <a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3" title="SUPINF_2:func.3">-</a> <span class="p2">(<span class="default"><span class="p3"><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="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:60"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="supinf_2.html#T16" title="SUPINF_2:th.16">SUPINF_2:16</a></i>;<br/>
<b>hence </b><a NAME="E8:60"/>
<span class="p1"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><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="supinf_2.html#K4" title="SUPINF_2:func.4">-</a> <span class="p3"><a href="extreal1.html#K3" title="EXTREAL1:func.3">|.</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span></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> <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#K4" title="SUPINF_2:func.4">-</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="extreal1.html#K3" title="EXTREAL1:func.3">.|</a></span>
 <b>by </b><i><a class="txt" href="#E2:60"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="extreal2.html#T60" target="_self" title="EXTREAL2:th.60">Th60</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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