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<div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</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="E1:26"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">c</font> &amp; <font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 ;<br/>

<a NAME="E2:26"/><i><font color="Green" title="E23">A2</font></i>: 
<span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">r</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E3:26_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:26_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="borsuk_5.html#T15" target="_self" title="BORSUK_5:th.15">Th15</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1"/>
( <font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> or  not <font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1"/>
<font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E3:26_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E3:26_1_1_1"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_2"/>
not <font color="Maroon" title="c5">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:26_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T15" target="_self" title="BORSUK_5:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E4:26_1_1_2"/>
<font color="Maroon" title="c4">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:26"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_2"/><i><font color="Green" title="E24">A5</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 
<b>by </b><i><a class="ref" href="limfunc1.html#T12" title="LIMFUNC1:th.12">LIMFUNC1:12</a></i>;<br/>
<a NAME="E2:26_2"/>
<span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:26"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="limfunc1.html#T9" title="LIMFUNC1:th.9">LIMFUNC1:9</a></i>;<br/>
<b>hence </b><a NAME="E3:26_2"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:26_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c3">c</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">[.</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="borsuk_5.html#NK3" title="BORSUK_5:NK.3">,+infty.[</a></span>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E2:26"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
