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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/>

<span class="kw">thus </span><a NAME="E1:108"/>
<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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
  <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</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="108_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:108_1"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/>

<a NAME="E2:108_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A2</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:108_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="108_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/><div class="add"><a NAME="E2:108_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></span>;<br/><a NAME="E3:108_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:108_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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/><div class="add"><a NAME="E2:108_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></span>;<br/><a NAME="E3:108_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:108_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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:108_1_1_3"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/><div class="add"><a NAME="E2:108_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="108_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:108_1_1_3_1_1"/>
(  not <font color="Maroon" title="c4">x</font> is  <a href="irrat_1.html#NV1" title="IRRAT_1:NV.1">irrational</a>  or <font color="Maroon" title="c4">x</font> is  <a href="irrat_1.html#NV1" title="IRRAT_1:NV.1">irrational</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_1_3_1_1_1"/>
not <font color="Maroon" title="c4">x</font> is  <a href="irrat_1.html#NV1" title="IRRAT_1:NV.1">irrational</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">hence </span><a NAME="E2:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E1:108_1_1_3"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="borsuk_5.html#T52" target="_self" title="BORSUK_5:th.52">Th52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_1_3_1_1_2"/>
<font color="Maroon" title="c4">x</font> is  <a href="irrat_1.html#NV1" title="IRRAT_1:NV.1">irrational</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">hence </span><a NAME="E2:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_1_1_3"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="borsuk_5.html#T53" target="_self" title="BORSUK_5:th.53">Th53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E3:108_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:108_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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:108_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span></span><br/><div class="add"><a NAME="E2:108_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T236" title="XXREAL_1:th.236">XXREAL_1:236</a></span>;<br/><span class="kw">hence </span><a NAME="E3:108_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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</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/>

<span class="kw">assume </span><a NAME="E2:108"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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>  ;<br/>
<a NAME="E4:108"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:108_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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:108"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_1"><span class="kw">suppose </span></a><a NAME="E1:108_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_1.html#K4" title="RCOMP_1:func.4">].</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_1.html#K4" title="RCOMP_1:func.4">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:108_2_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T234" title="XXREAL_1:th.234">XXREAL_1:234</a></span>;<br/><a NAME="E3:108_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A6</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T52" target="_self" title="BORSUK_5:th.52">Th52</a></span>;<br/><a NAME="E4:108_2_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_2_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_2"><span class="kw">suppose </span></a><a NAME="E1:108_2_2"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:108_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T84" target="_self" title="BORSUK_5:th.84">Th84</a></span>;<br/><a NAME="E3:108_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A8</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:108_2_2"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E4:108_2_2"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_2_2"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_2_3"><span class="kw">suppose </span></a><a NAME="E1:108_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_1.html#K3" title="RCOMP_1:func.3">[.</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_1.html#K3" title="RCOMP_1:func.3">.[</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:108_2_3"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T236" title="XXREAL_1:th.236">XXREAL_1:236</a></span>;<br/><a NAME="E3:108_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A9</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_5.html#T52" target="_self" title="BORSUK_5:th.52">Th52</a></span>;<br/><a NAME="E4:108_2_3"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:108_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_2_3"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
