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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be  <a href="xxreal_0.html#V1" title="XXREAL_0:attr.1">ext-real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> implies  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font> )</span><br/>







<a NAME="E1:32"/><i><font color="Green" title="E23">A1</font></i>: 
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" target="_self" title="XXREAL_0:th.17">Th17</a></i>;<br/>
<b>assume </b><a NAME="E2:32"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font></span><br/>

<a NAME="E3:32"/><b>then </b>
(  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> &amp;  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">d</font> )
 
<b>by </b><i><a class="txt" href="xxreal_0.html#E1:32"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" target="_self" title="XXREAL_0:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E4:32"/>
 <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D8" target="_self" title="XXREAL_0:def.8">Def8</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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