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<b>let </b><font color="Maroon" title="c1">r</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="E1:3"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/><font color="Olive" title="b1">r</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="axioms.html#T28" title="AXIOMS:th.28">AXIOMS:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">p</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <b> such that </b><br/><a NAME="E3:3"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E4:3"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T9" target="_self" title="SEQ_4:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:3"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c3">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E3:3"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">n1</font>
;<br/>

<b>thus </b><a NAME="E7:3"/>
<font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n1</font>
 <b>by </b><i><a class="txt" href="seq_4.html#E4:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="seq_4.html#E6:3"><i><font color="Green" title="E6">A3</font></i></a></i>;<br/>


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