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<b>let </b><font color="Maroon" title="c1">x</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="c1">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a> iff <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_5.html#K1" title="BORSUK_5:func.1">IRRAT</a>  )</span><br/>

<a NAME="E1:25"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_5.html#K1" title="BORSUK_5:func.1">IRRAT</a>  implies <font color="Maroon" title="c1">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:25_1"/>
<font color="Maroon" title="c1">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_5.html#K1" title="BORSUK_5:func.1">IRRAT</a> </span><br/><a NAME="E2:25_1"/><b>then </b>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 ;<br/><b>hence </b><a NAME="E3:25_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_5.html#K1" title="BORSUK_5:func.1">IRRAT</a> 
 <b>by </b><i><a class="txt" href="#E1:25"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:25"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_5.html#K1" title="BORSUK_5:func.1">IRRAT</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a></span><br/>

<a NAME="E4:25"/><b>then </b>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K3" title="NUMBERS:func.3">RAT</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:25"/>
<font color="Maroon" title="c1">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a>
 <b>by </b><i><a class="ref" href="rat_1.html#D2" title="RAT_1:def.2">RAT_1:def 2</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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