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<b>let </b><font color="Maroon" title="c2">r</font>, <font color="Maroon" title="c3">a</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 implies  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">a</font> )</span><br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 or  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<b>assume </b><a NAME="E2:4"/>
<font color="Maroon" title="c3">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">a</font></span><br/>

<a NAME="E3:4"/><b>then </b><i><font color="Green" title="E5">A2</font></i>: 
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span>
 
<b>by </b><i><a class="txt" href="topreala.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="topreala.html#T2" target="_self" title="TOPREALA:th.2">Th2</a></i>;<br/>
<a NAME="E4:4"/>
(  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> &amp;  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><font color="Maroon" title="c2">r</font></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> )
 
<b>by </b><i><a class="ref" href="int_1.html#D6" title="INT_1:def.6">INT_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:4"/>
 <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="int_1.html#K4" title="INT_1:func.4">frac</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="topreala.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="topreala.html#E3:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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