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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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/>

<a NAME="E1:19"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_5.html#NK2" title="BORSUK_5:NK.2">].-infty,</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="borsuk_5.html#NK2" title="BORSUK_5:NK.2">.[</a></span>
 ;<br/><a NAME="E2:19_1"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Olive" title="b1">g</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="prob_1.html#D15" title="PROB_1:def.15">PROB_1:def 15</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">h</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">h</font> )
 ;<br/><b>thus </b><a NAME="E5:19_1"/>
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E4:19_1"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:19"/>
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<a NAME="E4:19"/><b>then </b>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g</font> where <font color="Olive" title="b1">g</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">g</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:19"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_5.html#NK2" title="BORSUK_5:NK.2">].-infty,</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="borsuk_5.html#NK2" title="BORSUK_5:NK.2">.[</a></span>
 <b>by </b><i><a class="ref" href="prob_1.html#D15" title="PROB_1:def.15">PROB_1:def 15</a></i>;<br/>


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