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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">s</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>set </b><font color="Maroon">Q</font> = <a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K2">.[</a></span>;<br/>
<b>set </b><font color="Maroon">QQ</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r1</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:110_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K2">.[</a></span>
 ;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E3:110_1_1"/>
<a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:110_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> )
 ;<br/><a NAME="E6:110_1_1"/>
( <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> )
 <b>by </b><i><a class="ref" href="pscomp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>;<br/><b>hence </b><a NAME="E7:110_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r1</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:110_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r1</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:110_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> )
 ;<br/><b>set </b><font color="Maroon">p</font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span>;<br/><i><font color="Green">E34</font></i>: <a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="seq_1.html#K2">.</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">r2</font>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
;<br/><a NAME="E6:110_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:110_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:110"/>
<a href="pscomp_1.html#K14" target="_self">proj2</a>  <a href="relset_1.html#K11">"</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r1</font>,<font color="Olive">r2</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r1</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">r2</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r2</font> &amp; <font color="Olive">r2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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