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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/>

<b>assume </b><a NAME="E1:88"/><i><font color="Green" title="E58">A1</font></i>: 
<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/>

<b>assume </b><a NAME="E2:88"/>
 <a href="borsuk_5.html#K3" title="BORSUK_5:func.3">IRRAT</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="borsuk_5.html#K2" title="BORSUK_5:func.2">RAT</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:88"/><i><font color="Green" title="E59">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_5.html#K3" title="BORSUK_5:func.3">IRRAT</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="borsuk_5.html#K2" title="BORSUK_5:func.2">RAT</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as  <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>by </b><i><a class="txt" href="borsuk_5.html#E4:88"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:88"/>
( not <font color="Maroon" title="c4">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a> &amp; <font color="Maroon" title="c4">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a> )
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:88"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="borsuk_5.html#E4:88"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="borsuk_5.html#T52" target="_self" title="BORSUK_5:th.52">Th52</a>, <a class="ref" href="borsuk_5.html#T53" target="_self" title="BORSUK_5:th.53">Th53</a></i>;<br/>
<b>hence </b><a NAME="E7:88"/>
contradiction
 ;<br/>


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