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<b>let </b><font color="Maroon" title="c1">q1</font>, <font color="Maroon" title="c2">q2</font>, <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>







<b>thus </b><a NAME="E1:15"/>
<span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c2">q2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c2">q2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:15_1"/><i><font color="Green" title="E4">A2</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c2">q2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E1:15_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:15_1"/><b>then </b><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c1">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">q2</font> )
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T4" target="_self" title="RCOMP_2:th.4">Th4</a></i>;<br/>
<a NAME="E5:15_1"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E3:15_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="rcomp_2.html#T4" target="_self" title="RCOMP_2:th.4">Th4</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b><i><font color="Green" title="E7">A5</font></i>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x2</font>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E4:15_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="xxreal_0.html#T29" title="XXREAL_0:th.29">XXREAL_0:29</a></i>;<br/>
<a NAME="E7:15_1"/>
<font color="Maroon" title="c6">x2</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">q2</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E4:15_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="rcomp_2.html#E5:15_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="square_1.html#T39" title="SQUARE_1:th.39">SQUARE_1:39</a></i>;<br/>
<b>hence </b><a NAME="E8:15_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 <b>by </b><i><a class="txt" href="rcomp_2.html#E6:15_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green" title="E3">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c3">p1</font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c2">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E4:15"/><i><font color="Green" title="E4">A7</font></i>: 
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c6">x2</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">q2</font>,<font color="Maroon" title="c4">p2</font> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E2:15"><i><font color="Green" title="E3">A6</font></i></a>, <a class="ref" href="rcomp_2.html#T4" target="_self" title="RCOMP_2:th.4">Th4</a></i>;<br/>
<a NAME="E5:15"/><b>then </b><i><font color="Green" title="E5">A8</font></i>: 
( <font color="Maroon" title="c1">q1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x2</font> &amp; <font color="Maroon" title="c3">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">x2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T31" title="XXREAL_0:th.31">XXREAL_0:31</a></i>;<br/>
<a NAME="E6:15"/><i><font color="Green" title="E6">A9</font></i>: 
( <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">q2</font> &amp; <font color="Maroon" title="c6">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E4:15"><i><font color="Green" title="E4">A7</font></i></a>, <a class="ref" href="square_1.html#T39" title="SQUARE_1:th.39">SQUARE_1:39</a></i>;<br/>
<a NAME="E7:15"/><b>then </b><i><font color="Green" title="E7">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c2">q2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E5:15"><i><font color="Green" title="E5">A8</font></i></a></i>;<br/>
<a NAME="E8:15"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E5:15"><i><font color="Green" title="E5">A8</font></i></a>, <a class="txt" href="rcomp_2.html#E6:15"><i><font color="Green" title="E6">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:15"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c1">q1</font>,<font color="Maroon" title="c2">q2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 <b>by </b><i><a class="txt" href="rcomp_2.html#E7:15"><i><font color="Green" title="E7">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div>
