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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="membered.html#V2" title="MEMBERED:attr.2">ext-real-membered</a> <a href="xxreal_2.html#V1" title="XXREAL_2:attr.1">left_end</a> <a href="xxreal_2.html#V2" title="XXREAL_2:attr.2">right_end</a> <a href="xxreal_2.html#V6" title="XXREAL_2:attr.6">connected</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/><b>let </b><font color="Maroon" title="c2">x</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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="membered.html#D14" title="MEMBERED:def.14">MEMBERED:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> ) &amp; (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> ) )</span><br/>



<b>thus </b><a NAME="E1:126"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="126_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:126_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span></span><br/>

<a NAME="E2:126_1"/><b>then </b>
(  <a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <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">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="xxreal_2.html#T3" target="_self" title="XXREAL_2:th.3">Th3</a>, <a class="ref" href="xxreal_2.html#T4" target="_self" title="XXREAL_2:th.4">Th3B</a></i>;<br/>
<b>hence </b><a NAME="E3:126_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<a NAME="E2:126"/>
(  <a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D5" target="_self" title="XXREAL_2:def.5">Def5</a>, <a class="ref" href="xxreal_2.html#D6" target="_self" title="XXREAL_2:def.6">Def6</a></i>;<br/>
<a NAME="E3:126"/><b>then </b>
<span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="xxreal_2.html#D12" target="_self" title="XXREAL_2:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E4:126"/>
(  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_2.html#NK3" title="XXREAL_2:NK.3">min</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_2.html#NK4" title="XXREAL_2:NK.4">max</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="xxreal_1.html#K1" title="XXREAL_1:func.1">.]</a></span> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> )
 ; <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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