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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><b>that </b><br/><a NAME="E1:54"/><i><font color="Green" title="E23">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>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a> <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#NK2" title="BORSUK_5:NK.2">].-infty,</a> <font color="Maroon" title="c1">a</font>
 ;<br/>

<b>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="E4:54"/><i><font color="Green" title="E25">A3</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#NK3" title="BORSUK_5:NK.3">[.</a> <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#NK2" title="BORSUK_5:NK.2">].-infty,</a> <font color="Maroon" title="c1">a</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a>, <a class="txt" href="isomichi.html#E2:54"><i><font color="Green" title="E24">A2</font></i></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="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a>, <a class="txt" href="isomichi.html#E4:54"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<a NAME="E6:54"/>
( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">a</font> )
 
<b>by </b><i><a class="txt" href="isomichi.html#E4:54"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="borsuk_5.html#T15" title="BORSUK_5:th.15">BORSUK_5:15</a>, <a class="ref" href="borsuk_5.html#T17" title="BORSUK_5:th.17">BORSUK_5:17</a></i>;<br/>
<b>hence </b><a NAME="E7:54"/>
contradiction
 <b>by </b><i><a class="txt" href="isomichi.html#E1:54"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


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