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

<a NAME="E1:17"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 ;<br/><a NAME="E2:17_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#D2">LIMFUNC1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:17_1"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>thus </b><a NAME="E5:17_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E4:17_1"><i><font color="Green">E12</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:17"/>
<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:17"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:17"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:17"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" target="_self">[.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="borsuk_5.html#NK3" target="_self">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="limfunc1.html#D2">LIMFUNC1:def 2</a></i>;<br/>


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