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<div class="add">

<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1"/>
( <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> or <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><b>suppose </b></a><a NAME="E1:91_1_1"/><i><font color="Green" title="E61">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/><div class="add"><b>thus </b><a NAME="E2:91_1_1"/>
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</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> <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )</span><br/>

<b>assume </b><a NAME="E1:91_1_1_1"/><i><font color="Green" title="E62">A2</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/>

<a NAME="E2:91_1_1_1"/><b>then </b><i><font color="Green" title="E63">A3</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; not <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#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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#E1:91_1_1_1"><i><font color="Green" title="E62">A2</font></i></a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1"/>
( ( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</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="c2">b</font> ) or ( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <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> ) or ( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</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="c2">b</font> ) or ( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <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> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</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="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/><div class="add"><a NAME="E2:91_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self" title="BORSUK_5:th.16">Th16</a></i>;<br/><a NAME="E3:91_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</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/><b>hence </b><a NAME="E4:91_1_1_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</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>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font> &amp; <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/><div class="add"><a NAME="E2:91_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self" title="BORSUK_5:th.16">Th16</a></i>;<br/><a NAME="E3:91_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</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/><b>hence </b><a NAME="E4:91_1_1_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</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>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_3"/><i><font color="Green" title="E64">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</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="c2">b</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/><div class="add"><a NAME="E2:91_1_1_1_1_3"/>
<font color="Maroon" title="c4">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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_1_1_3_1_1"/>
(  not <font color="Maroon" title="c4">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a> or <font color="Maroon" title="c4">x</font> is <a href="rat_1.html#V1" title="RAT_1:attr.1">irrational</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_3_1_1_1"/>
not <font color="Maroon" title="c4">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="c4">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></span><br/><div class="add"><b>hence </b><a NAME="E2:91_1_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c4">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>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="borsuk_5.html#E2:91_1_1_1"><i><font color="Green" title="E63">A3</font></i></a>, <a class="txt" href="borsuk_5.html#E1:91_1_1_1_1_3"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="borsuk_5.html#T52" target="_self" title="BORSUK_5:th.52">Th52</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_3_1_1_2"/>
<font color="Maroon" title="c4">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="c4">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></span><br/><div class="add"><b>hence </b><a NAME="E2:91_1_1_1_1_3_1_1_2"/>
<font color="Maroon" title="c4">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>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green" title="E61">A1</font></i></a>, <a class="txt" href="borsuk_5.html#E1:91_1_1_1_1_3"><i><font color="Green" title="E64">A4</font></i></a>, <a class="ref" href="borsuk_5.html#T53" target="_self" title="BORSUK_5:th.53">Th53</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></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E3:91_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</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/><b>hence </b><a NAME="E4:91_1_1_1_1_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</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>; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:91_1_1_1_1_4"/>
( <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">a</font> &amp; <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/><div class="add"><a NAME="E2:91_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</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="E3:91_1_1_1_1_4"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</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>; <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></div><b>end;</b></div>

</div><b>end;</b></div><b>let </b><font color="Maroon" title="c3">x</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> <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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> )</span><br/><b>assume </b><a NAME="E3:91_1_1"/><i><font color="Green" title="E62">A5</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/><b>then 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></i>;<br/><a NAME="E5:91_1_1"/><i><font color="Green" title="E63">A6</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 <b>by </b><i><a class="txt" href="borsuk_5.html#E3:91_1_1"><i><font color="Green" title="E62">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> or <font color="Maroon" title="c4">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> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> )
 </span><b>by </b><i><a class="txt" href="borsuk_5.html#E5:91_1_1"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2_1"><b>suppose </b></a><a NAME="E1:91_1_1_2_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/><div class="add"><a NAME="E2:91_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T16" target="_self" title="BORSUK_5:th.16">Th16</a></i>;<br/><a NAME="E3:91_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; not <font color="Maroon" title="c4">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="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T52" target="_self" title="BORSUK_5:th.52">Th52</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2_2"><b>suppose </b></a><a NAME="E1:91_1_1_2_2"/><i><font color="Green" title="E64">A7</font></i>: 
<font color="Maroon" title="c4">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/><div class="add"><a NAME="E2:91_1_1_2_2"/><i><font color="Green" title="E65">A8</font></i>: 
<font color="Maroon" title="c4">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/><a NAME="E3:91_1_1_2_2"/>
 <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#R1" title="XBOOLE_0:pred.1">misses</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="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T84" target="_self" title="BORSUK_5:th.84">Th84</a></i>;<br/><a NAME="E4:91_1_1_2_2"/><b>then </b>
not <font color="Maroon" title="c4">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="txt" href="borsuk_5.html#E1:91_1_1_2_2"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E5:91_1_1_2_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_5.html#E2:91_1_1_2_2"><i><font color="Green" title="E65">A8</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2_3"><b>suppose </b></a><a NAME="E1:91_1_1_2_3"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/><div class="add"><a NAME="E2:91_1_1_2_3"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="borsuk_5.html#T15" target="_self" title="BORSUK_5:th.15">Th15</a></i>;<br/><a NAME="E3:91_1_1_2_3"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; not <font color="Maroon" title="c4">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="txt" href="borsuk_5.html#E1:91_1_1"><i><font color="Green" title="E61">A1</font></i></a>, <a class="ref" href="borsuk_5.html#T52" target="_self" title="BORSUK_5:th.52">Th52</a>, <a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1_2_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</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></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_2"><b>suppose </b></a><a NAME="E1:91_1_2"/><i><font color="Green" title="E61">A9</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span><br/><div class="add"><a NAME="E2:91_1_2"/><b>then </b><i><font color="Green" title="E62">A10</font></i>: 
<span class="p1"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <b>by </b><i><a class="txt" href="borsuk_5.html#E100"><i><font color="Green" title="E59">Lm11</font></i></a></i>;<br/><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> = 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:91_1_2"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="borsuk_5.html#E99"><i><font color="Green" title="E58">Lm10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_5.html#E2:91_1_2"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
;<br/><b>hence </b><a NAME="E4:91_1_2"/>
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">].</a><span class="default"><a href="xxreal_0.html#K2" title="XXREAL_0:func.2">-infty</a> ,<font color="Maroon" title="c1">a</font></span><a href="rcomp_2.html#K2" title="RCOMP_2:func.2">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><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></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">[.</a><span class="default"><font color="Maroon" title="c2">b</font>,<a href="xxreal_0.html#K1" title="XXREAL_0:func.1">+infty</a> </span><a href="rcomp_2.html#K1" title="RCOMP_2:func.1">.[</a></span>
 ; <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></div><b>end;</b></div>

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