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<b>consider </b><font color="Maroon" title="c1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span>;<br/>
<b>assume </b><a NAME="E2:31"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:31"/><b>then </b><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>,<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default">1,1</span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
 ex <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="topreal1.html#T19" target="_self" title="TOPREAL1:th.19">Th19</a></i>;<br/>
<a NAME="E5:31"/>
 ex <font color="Olive" title="b1">p2</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">p2</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="topreal1.html#E3:31"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="topreal1.html#T19" target="_self" title="TOPREAL1:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E6:31"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal1.html#E4:31"><i><font color="Green" title="E19">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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