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<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">g</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">g</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  holds <br/>for <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/> ex <font color="Olive" title="b3">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">g</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  implies for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/> ex <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">s</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:16"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">g</font>
 <span class="kw">and </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
 <span class="kw">and </span><br/><a NAME="E3:16"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/> ex <font color="Olive" title="b2">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">s</font> )</span><br/>

<a NAME="E4:16"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="rcomp_1.html#V1" title="RCOMP_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T24" title="RCOMP_1:th.24">RCOMP_1:24</a></span>;<br/>
<a NAME="E5:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="xxreal_2.html#V5" title="XXREAL_2:attr.5">bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="fcont_1.html#T30" title="FCONT_1:th.30">FCONT_1:30</a>, <a class="ref" href="rcomp_1.html#T28" title="RCOMP_1:th.28">RCOMP_1:28</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">ub</font> =  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">lb</font> =  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>;<br/>
<a NAME="E6:16"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="xxreal_1.html#T1" title="XXREAL_1:th.1">XXREAL_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x2</font>, <font color="Maroon" title="c7">x1</font> being   <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"> such that </span><br/><a NAME="E8:16"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c6">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E9:16"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:16"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:16"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="fcont_1.html#T32" title="FCONT_1:th.32">FCONT_1:32</a>, <a class="ref" href="rcomp_1.html#T24" title="RCOMP_1:th.24">RCOMP_1:24</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c6">x2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.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">let </span><font color="Maroon" title="c10">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> implies  ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">s</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E13:16"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">s</font> )</span><br/>

<a NAME="E14:16"/>
<font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E15:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K4" title="SEQ_4:func.4">upper_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p4"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span></span>)</span></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="seq_4.html#T24" title="SEQ_4:th.24">SEQ_4:24</a>, <a class="ref" href="xxreal_1.html#T222" title="XXREAL_1:th.222">XXREAL_1:222</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_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="E1:16_1_1"/>
( <font color="Maroon" title="c8">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x2</font> or <font color="Maroon" title="c9">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_1"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c8">x1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">s</font> )</span><br/><div class="add"><a NAME="E2:16_1_1_1"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/><a NAME="E3:16_1_1_1"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/><a NAME="E4:16_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:16_1_1_1"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">x2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:16_1_1_1"/><span class="lab"><font color="Green" title="E24">A15</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E1:16_1_1_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E2:16_1_1_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="fcont_2.html#T16" target="_self" title="FCONT_2:th.16">Th16</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">s</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="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">s</font> )</span><br/><span class="kw">thus </span><a NAME="E8:16_1_1_1"/>
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E6:16_1_1_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="txt" href="#E7:16_1_1_1"><span class="lab"><font color="Green" title="E24">A15</font></span></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="16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:16_1_1_2"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<font color="Maroon" title="c9">x2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">x1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">s</font> )</span><br/><div class="add"><a NAME="E2:16_1_1_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c8">x1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/><a NAME="E3:16_1_1_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c8">x1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E9:16"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="xxreal_2.html#D12" title="XXREAL_2:def.12">XXREAL_2:def 12</a></span>;<br/><a NAME="E4:16_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c8">x1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E6:16_1_1_2"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c8">x1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:16_1_1_2"/><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E13:16"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E10:16"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E1:16_1_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E2:16_1_1_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="ref" href="fcont_2.html#T16" target="_self" title="FCONT_2:th.16">Th16</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c12">s</font> = <font color="Maroon" title="c11">s</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="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">s</font> )</span><br/><span class="kw">thus </span><a NAME="E8:16_1_1_2"/>
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c12">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="#E6:16_1_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="#E7:16_1_1_2"><span class="lab"><font color="Green" title="E24">A20</font></span></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">hence </span><a NAME="E17:16"/>
 ex <font color="Olive" title="b1">s</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">g</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c10">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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