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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>thus </b><a NAME="E1:14"/>
<span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_2.html#K1" target="_self">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:14_1"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E1:14_1"><i><font color="Green">E3</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:14_1"/><b>then </b><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:14_1"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E3:14_1"><i><font color="Green">E4</font></i></a>, <a class="ref" href="rcomp_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E6:14_1"/><b>then </b><i><font color="Green">E7</font></i>: 
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E4:14_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<a NAME="E7:14_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E4:14_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rcomp_2.html#E5:14_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="square_1.html#T113">SQUARE_1:113</a></i>;<br/>
<b>hence </b><a NAME="E8:14_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_2.html#K1" target="_self">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_2.html#E6:14_1"><i><font color="Green">E7</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:14"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="rcomp_2.html#K1" target="_self">.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:14"/><i><font color="Green">E4</font></i>: 
(  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E2:14"><i><font color="Green">E3</font></i></a>, <a class="ref" href="rcomp_2.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E5:14"/><b>then </b><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<a NAME="E6:14"/><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E4:14"><i><font color="Green">E4</font></i></a>, <a class="ref" href="square_1.html#T113">SQUARE_1:113</a></i>;<br/>
<a NAME="E7:14"/><b>then </b><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E5:14"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E8:14"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span>
 
<b>by </b><i><a class="txt" href="rcomp_2.html#E5:14"><i><font color="Green">E5</font></i></a>, <a class="txt" href="rcomp_2.html#E6:14"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:14"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="rcomp_2.html#K1" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="rcomp_2.html#K1" target="_self">.[</a></span>
 <b>by </b><i><a class="txt" href="rcomp_2.html#E7:14"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


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