<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</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> ;<br/><b>let </b><font color="Maroon" title="c3">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>





<b>given </b><font color="Maroon" title="c4">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><a NAME="E1:7"/><i><font color="Green" title="E4">A1</font></i>: 
( <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; <font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c1">f</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font> &amp; <font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c3">x0</font> )
 ;<br/>

<a NAME="E2:7"/><i><font color="Green" title="E5">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> holds <br/> ex <font color="Olive" title="b2">c</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Olive" title="b1">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b2">c</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E6">A3</font></i>: 
( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> )
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> ) </span>}</span> 
 
;<br/>
<a NAME="E3:7_1"/><b>then </b><i><font color="Green" title="E7">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/>
<a NAME="E4:7_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T33" title="FDIFF_1:th.33">FDIFF_1:33</a></i>;<br/>
<a NAME="E5:7_1"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T33" title="FDIFF_1:th.33">FDIFF_1:33</a></i>;<br/>
<a NAME="E6:7_1"/><i><font color="Green" title="E10">A7</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/>

<a NAME="E2:7_1_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g2</font> where <font color="Olive" title="b1">g2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g2</font> &amp; <font color="Olive" title="b1">g2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">g2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:7_1_1"/><i><font color="Green" title="E11">A8</font></i>: 
( <font color="Maroon" title="c7">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">g2</font> &amp; <font color="Maroon" title="c7">g2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font> )
 ;<br/>
<a NAME="E5:7_1_1"/>
<font color="Maroon" title="c7">g2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="l_hospit.html#E4:7_1_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:7_1_1"/><b>then </b>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g3</font> where <font color="Olive" title="b1">g3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g3</font> &amp; <font color="Olive" title="b1">g3</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="l_hospit.html#E4:7_1_1"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:7_1"/><b>then </b><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/>
<a NAME="E8:7_1"/><i><font color="Green" title="E12">A10</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a></i>;<br/>
<a NAME="E9:7_1"/>
<font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E3:7_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E10:7_1"/><b>then </b><i><font color="Green" title="E13">A11</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T32" title="FDIFF_1:th.32">FDIFF_1:32</a></i>;<br/>
<a NAME="E11:7_1"/>
<font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E3:7_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E12:7_1"/><b>then </b><i><font color="Green" title="E14">A12</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c5">x</font>
 
<b>by </b><i><a class="ref" href="fdiff_1.html#T32" title="FDIFF_1:th.32">FDIFF_1:32</a></i>;<br/>
<a NAME="E13:7_1"/><i><font color="Green" title="E15">A13</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_2"/><i><font color="Green" title="E16">A14</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_2_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E4:7_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="fcont_1.html#T14" title="FCONT_1:th.14">FCONT_1:14</a></i>;<br/>
<a NAME="E2:7_1_2_1"/><b>then </b><i><font color="Green" title="E17">A15</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:7_1_2_1"/><i><font color="Green" title="E18">A16</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fcont_1.html#T2" title="FCONT_1:th.2">FCONT_1:2</a></i>;<br/>
<a NAME="E4:7_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E10:7_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="ref" href="fcont_1.html#T2" title="FCONT_1:th.2">FCONT_1:2</a></i>;<br/>
<a NAME="E5:7_1_2_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E3:7_1_2_1"><i><font color="Green" title="E18">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E6:7_1_2_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E2:7_1_2_1"><i><font color="Green" title="E17">A15</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_2_1"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="rcomp_1.html#T11" title="RCOMP_1:th.11">RCOMP_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:7_1_2"/>
for <font color="Olive" title="b1">s</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s</font> <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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b1">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/>( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">s</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:7_1_2_2"/><i><font color="Green" title="E17">A17</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s</font> <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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">z</font> =  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font>;<br/>
<a NAME="E2:7_1_2_2"/><i><font color="Green" title="E18">A18</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:7_1_2_2"/><i><font color="Green" title="E19">A19</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="ref" href="rcomp_1.html#T11" title="RCOMP_1:th.11">RCOMP_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_2_2_1_1"/>
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><b>by </b><i><a class="txt" href="l_hospit.html#E3:7_1_2_2"><i><font color="Green" title="E19">A19</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_2_2_1_1_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:7_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/><a NAME="E3:7_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">f</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T32" title="FDIFF_1:th.32">FDIFF_1:32</a></i>;<br/><b>hence </b><a NAME="E4:7_1_2_2_1_1_1"/>
( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1_2_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="l_hospit.html#E2:7_1_2_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_2_2_1_1_2"/><i><font color="Green" title="E20">A20</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_2_2_1_1_2_1_1"/>
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 </span><b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1_2_2_1_1_2"><i><font color="Green" title="E20">A20</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_2_2_1_1_2_1_1_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="l_hospit.html#E2:7_1_2_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_2_2_1_1_2_1_1_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_2_1_1_2_1_1_2"/>
( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E10:7_1"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_2"><i><font color="Green" title="E17">A17</font></i></a>, <a class="txt" href="l_hospit.html#E2:7_1_2_2"><i><font color="Green" title="E18">A18</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:7_1_2_2_1_1_2"/>
( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:7_1_2_2"/>
( <font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_2"/>
<font color="Maroon" title="c1">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1_2"><i><font color="Green" title="E16">A14</font></i></a>, <a class="ref" href="fcont_1.html#T14" title="FCONT_1:th.14">FCONT_1:14</a></i>;<br/>

<a NAME="E4:7_1_2"/><i><font color="Green" title="E17">A21</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_2_3"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E5:7_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="fcont_1.html#T14" title="FCONT_1:th.14">FCONT_1:14</a></i>;<br/>
<a NAME="E2:7_1_2_3"/><b>then </b><i><font color="Green" title="E18">A22</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:7_1_2_3"/><i><font color="Green" title="E19">A23</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fcont_1.html#T2" title="FCONT_1:th.2">FCONT_1:2</a></i>;<br/>
<a NAME="E4:7_1_2_3"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E12:7_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="ref" href="fcont_1.html#T2" title="FCONT_1:th.2">FCONT_1:2</a></i>;<br/>
<a NAME="E5:7_1_2_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E3:7_1_2_3"><i><font color="Green" title="E19">A23</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" title="ZFMISC_1:th.38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E6:7_1_2_3"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E2:7_1_2_3"><i><font color="Green" title="E18">A22</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_2_3"/>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="rcomp_1.html#T11" title="RCOMP_1:th.11">RCOMP_1:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:7_1_2"/>
for <font color="Olive" title="b1">s</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">s</font> <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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Olive" title="b1">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/>( <font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">s</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:7_1_2_4"/><i><font color="Green" title="E18">A24</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s</font> <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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 ;<br/>

<b>set </b><font color="Maroon" title="c7">z</font> =  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font>;<br/>
<a NAME="E2:7_1_2_4"/><i><font color="Green" title="E19">A25</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E4:7_1_2"><i><font color="Green" title="E17">A21</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_4"><i><font color="Green" title="E18">A24</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:7_1_2_4"/><i><font color="Green" title="E20">A26</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_4"><i><font color="Green" title="E18">A24</font></i></a>, <a class="ref" href="rcomp_1.html#T11" title="RCOMP_1:th.11">RCOMP_1:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_2_4_1_1"/>
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 </span><b>by </b><i><a class="txt" href="l_hospit.html#E3:7_1_2_4"><i><font color="Green" title="E20">A26</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_2_4_1_1_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ;<br/><div class="add"><a NAME="E2:7_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/><a NAME="E3:7_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font>
 <b>by </b><i><a class="ref" href="fdiff_1.html#T32" title="FDIFF_1:th.32">FDIFF_1:32</a></i>;<br/><b>hence </b><a NAME="E4:7_1_2_4_1_1_1"/>
( <font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1_2_4"><i><font color="Green" title="E18">A24</font></i></a>, <a class="txt" href="l_hospit.html#E2:7_1_2_4"><i><font color="Green" title="E19">A25</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_2_4_1_1_2"/><i><font color="Green" title="E21">A27</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_4_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_4_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_2_4_1_1_2_1_1"/>
(  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> or  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 </span><b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1_2_4_1_1_2"><i><font color="Green" title="E21">A27</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_4_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:7_1_2_4_1_1_2_1_1_1"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_4_1_1_2_1_1_1"/>
( <font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_4"><i><font color="Green" title="E18">A24</font></i></a>, <a class="txt" href="l_hospit.html#E2:7_1_2_4"><i><font color="Green" title="E19">A25</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_4_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:7_1_2_4_1_1_2_1_1_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_4_1_1_2_1_1_2"/>
( <font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E12:7_1"><i><font color="Green" title="E14">A12</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_1_2_4"><i><font color="Green" title="E18">A24</font></i></a>, <a class="txt" href="l_hospit.html#E2:7_1_2_4"><i><font color="Green" title="E19">A25</font></i></a>, <a class="ref" href="fcont_1.html#D1" title="FCONT_1:def.1">FCONT_1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:7_1_2_4_1_1_2"/>
( <font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:7_1_2_4"/>
( <font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c6">s</font></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:7_1_2"/>
<font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="txt" href="l_hospit.html#E4:7_1_2"><i><font color="Green" title="E17">A21</font></i></a>, <a class="ref" href="fcont_1.html#T14" title="FCONT_1:th.14">FCONT_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:7_1"/><b>then </b><i><font color="Green" title="E16">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="fcont_1.html#T21" title="FCONT_1:th.21">FCONT_1:21</a></i>;<br/>
<a NAME="E15:7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E13:7_1"><i><font color="Green" title="E15">A13</font></i></a>, <a class="ref" href="fcont_1.html#T21" title="FCONT_1:th.21">FCONT_1:21</a></i>;<br/>
<a NAME="E16:7_1"/><b>then </b><i><font color="Green" title="E17">A29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E14:7_1"><i><font color="Green" title="E16">A28</font></i></a>, <a class="ref" href="fcont_1.html#T19" title="FCONT_1:th.19">FCONT_1:19</a></i>;<br/>
<a NAME="E17:7_1"/><i><font color="Green" title="E18">A30</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E18:7_1"/><i><font color="Green" title="E19">A31</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font>
 
<b>by </b><i><a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>;<br/>
<a NAME="E19:7_1"/><i><font color="Green" title="E20">A32</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E4:7_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="l_hospit.html#E17:7_1"><i><font color="Green" title="E18">A30</font></i></a>, <a class="ref" href="fcont_1.html#T14" title="FCONT_1:th.14">FCONT_1:14</a></i>;<br/>
<a NAME="E20:7_1"/><i><font color="Green" title="E21">A33</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E5:7_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="l_hospit.html#E18:7_1"><i><font color="Green" title="E19">A31</font></i></a>, <a class="ref" href="fcont_1.html#T14" title="FCONT_1:th.14">FCONT_1:14</a></i>;<br/>
<a NAME="E21:7_1"/><i><font color="Green" title="E22">A34</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="l_hospit.html#E19:7_1"><i><font color="Green" title="E20">A32</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E22:7_1"/><b>then </b><i><font color="Green" title="E23">A35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E7:7_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></i>;<br/>
<a NAME="E23:7_1"/><i><font color="Green" title="E24">A36</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="l_hospit.html#E20:7_1"><i><font color="Green" title="E21">A33</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E24:7_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E21:7_1"><i><font color="Green" title="E22">A34</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E25:7_1"/><b>then </b><i><font color="Green" title="E25">A37</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E26:7_1"/><i><font color="Green" title="E26">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E8:7_1"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="l_hospit.html#E23:7_1"><i><font color="Green" title="E24">A36</font></i></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></i>;<br/>
<a NAME="E27:7_1"/><b>then </b><i><font color="Green" title="E27">A39</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E22:7_1"><i><font color="Green" title="E23">A35</font></i></a>, <a class="txt" href="l_hospit.html#E25:7_1"><i><font color="Green" title="E25">A37</font></i></a>, <a class="ref" href="fdiff_1.html#T27" title="FDIFF_1:th.27">FDIFF_1:27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">f1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>;<br/>
<a NAME="E28:7_1"/><i><font color="Green" title="E28">A40</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E16:7_1"><i><font color="Green" title="E17">A29</font></i></a>, <a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a></i>;<br/>
<a NAME="E29:7_1"/><i><font color="Green" title="E29">A41</font></i>: 
( <font color="Maroon" title="c3">x0</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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; <font color="Maroon" title="c5">x</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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E30:7_1"/><b>then </b>
( <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E28:7_1"><i><font color="Green" title="E28">A40</font></i></a></i>;<br/>
<a NAME="E31:7_1"/><b>then </b>
( <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/>
<a NAME="E32:7_1"/><b>then </b><i><font color="Green" title="E30">A42</font></i>: 
( <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> &amp; <font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</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/>
<i><font color="Green" title="E31">A43</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="l_hospit.html#E28:7_1"><i><font color="Green" title="E28">A40</font></i></a>, <a class="txt" href="l_hospit.html#E29:7_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c3">x0</font></span>)</span>
<b>by </b><i><a class="txt" href="l_hospit.html#E32:7_1"><i><font color="Green" title="E30">A42</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
0 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0</span>)</span>
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E32:7_1"><i><font color="Green" title="E30">A42</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
0

;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="l_hospit.html#E28:7_1"><i><font color="Green" title="E28">A40</font></i></a>, <a class="txt" href="l_hospit.html#E29:7_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="l_hospit.html#E32:7_1"><i><font color="Green" title="E30">A42</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="l_hospit.html#E32:7_1"><i><font color="Green" title="E30">A42</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
0

;<br/>
<b>then consider </b><font color="Maroon" title="c7">t</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E36:7_1"/><i><font color="Green" title="E32">A44</font></i>: 
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K7" title="SEQ_1:func.7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="l_hospit.html#E16:7_1"><i><font color="Green" title="E17">A29</font></i></a>, <a class="txt" href="l_hospit.html#E27:7_1"><i><font color="Green" title="E27">A39</font></i></a>, <a class="txt" href="l_hospit.html#E33:7_1"><i><font color="Green" title="E31">A43</font></i></a>, <a class="ref" href="rolle.html#T1" title="ROLLE:th.1">ROLLE:1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">t</font>
;<br/>

<a NAME="E37:7_1"/><i><font color="Green" title="E33">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E26:7_1"><i><font color="Green" title="E26">A38</font></i></a>, <a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E38:7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E22:7_1"><i><font color="Green" title="E23">A35</font></i></a>, <a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E39:7_1"/><b>then </b><i><font color="Green" title="E34">A46</font></i>: 
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c1">f</font></span>)</span>,<font color="Maroon" title="c7">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="txt" href="l_hospit.html#E37:7_1"><i><font color="Green" title="E33">A45</font></i></a>, <a class="ref" href="fdiff_1.html#T22" title="FDIFF_1:th.22">FDIFF_1:22</a></i>;<br/>
<a NAME="E40:7_1"/><i><font color="Green" title="E35">A47</font></i>: 
<font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E41:7_1"/>
<font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Maroon" title="c7">t</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="fdiff_1.html#T16" title="FDIFF_1:th.16">FDIFF_1:16</a></i>;<br/>
<a NAME="E42:7_1"/><b>then </b>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="seq_1.html#K13" title="SEQ_1:func.13">(#)</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E39:7_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="ref" href="fdiff_1.html#T23" title="FDIFF_1:th.23">FDIFF_1:23</a></i>;<br/>
<a NAME="E43:7_1"/><b>then </b>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E40:7_1"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="fdiff_1.html#T23" title="FDIFF_1:th.23">FDIFF_1:23</a></i>;<br/>
<a NAME="E44:7_1"/><b>then </b><i><font color="Green" title="E36">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">t</font></span>)</span>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1_5"/>
<font color="Maroon" title="c8">y</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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:7_1_5"/><b>then </b><i><font color="Green" title="E37">A49</font></i>: 
( <font color="Maroon" title="c8">y</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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:7_1_5"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g4</font> where <font color="Olive" title="b1">g4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g4</font> &amp; <font color="Olive" title="b1">g4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">g4</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:7_1_5"/><i><font color="Green" title="E38">A50</font></i>: 
( <font color="Maroon" title="c9">g4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">g4</font> &amp; <font color="Maroon" title="c9">g4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> )
 ;<br/><a NAME="E6:7_1_5"/><i><font color="Green" title="E39">A51</font></i>: 
<font color="Maroon" title="c9">g4</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="l_hospit.html#E5:7_1_5"><i><font color="Green" title="E38">A50</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:7_1_5"/>
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g4</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E2:7_1_5"><i><font color="Green" title="E37">A49</font></i></a>, <a class="txt" href="l_hospit.html#E5:7_1_5"><i><font color="Green" title="E38">A50</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E8:7_1_5"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">g4</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E5:7_1_5"><i><font color="Green" title="E38">A50</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E9:7_1_5"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g5</font> where <font color="Olive" title="b1">g5</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g5</font> &amp; <font color="Olive" title="b1">g5</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="l_hospit.html#E5:7_1_5"><i><font color="Green" title="E38">A50</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1_5"><i><font color="Green" title="E39">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E10:7_1_5"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E46:7_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E47:7_1"/><b>then </b><i><font color="Green" title="E37">A52</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E48:7_1"/><b>then </b><i><font color="Green" title="E38">A53</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>;<br/>
<a NAME="E49:7_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E50:7_1"/><b>then </b><i><font color="Green" title="E39">A54</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E48:7_1"><i><font color="Green" title="E38">A53</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E51:7_1"/>
not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E52:7_1"/><b>then </b><i><font color="Green" title="E40">A55</font></i>: 
<font color="Maroon" title="c5">x</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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">x0</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E29:7_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E53:7_1"/><b>then </b><i><font color="Green" title="E41">A56</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">g</font> &amp; not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E50:7_1"><i><font color="Green" title="E39">A54</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<div><i><font color="Green" title="E42">A57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_6"/>
<font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:7_1_6"/><b>then </b>
<font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:7_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="l_hospit.html#E53:7_1"><i><font color="Green" title="E41">A56</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E55:7_1"/><i><font color="Green" title="E43">A58</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E56:7_1"/><i><font color="Green" title="E44">A59</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</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="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E57:7_1"/>
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g6</font> where <font color="Olive" title="b1">g6</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g6</font> &amp; <font color="Olive" title="b1">g6</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="l_hospit.html#E55:7_1"><i><font color="Green" title="E43">A58</font></i></a></i>;<br/>
<a NAME="E58:7_1"/><b>then </b>
<font color="Maroon" title="c3">x0</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="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E59:7_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E3:7_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="l_hospit.html#E56:7_1"><i><font color="Green" title="E44">A59</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E60:7_1"/><b>then </b><i><font color="Green" title="E45">A60</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E61:7_1"/><b>then </b><i><font color="Green" title="E46">A61</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>;<br/>
<a NAME="E62:7_1"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E63:7_1"/><b>then </b><i><font color="Green" title="E47">A62</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</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">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E61:7_1"><i><font color="Green" title="E46">A61</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E64:7_1"/>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E65:7_1"/><b>then </b><i><font color="Green" title="E48">A63</font></i>: 
<font color="Maroon" title="c7">t</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="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a></i>;<br/>
<a NAME="E66:7_1"/><b>then </b><i><font color="Green" title="E49">A64</font></i>: 
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> &amp; not <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E63:7_1"><i><font color="Green" title="E47">A62</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_7"/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><a NAME="E2:7_1_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/><a NAME="E3:7_1_7"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:7_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="l_hospit.html#E66:7_1"><i><font color="Green" title="E49">A64</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E68:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E44:7_1"><i><font color="Green" title="E36">A48</font></i></a>, <a class="txt" href="l_hospit.html#E54:7_1"><i><font color="Green" title="E42">A57</font></i></a>, <a class="ref" href="xcmplx_1.html#T95" title="XCMPLX_1:th.95">XCMPLX_1:95</a></i>;<br/>
<a NAME="E69:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E70:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>;<br/>
<a NAME="E71:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E47:7_1"><i><font color="Green" title="E37">A52</font></i></a>, <a class="txt" href="l_hospit.html#E52:7_1"><i><font color="Green" title="E40">A55</font></i></a>, <a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>;<br/>
<a NAME="E72:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<a NAME="E73:7_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E6:7_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E74:7_1"/>
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<font color="Maroon" title="c5">x</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c7">t</font> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E36:7_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="txt" href="l_hospit.html#E60:7_1"><i><font color="Green" title="E45">A60</font></i></a>, <a class="txt" href="l_hospit.html#E65:7_1"><i><font color="Green" title="E48">A63</font></i></a>, <a class="ref" href="rfunct_1.html#D4" title="RFUNCT_1:def.4">RFUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:7"/><i><font color="Green" title="E6">A65</font></i>: 
for <font color="Olive" title="b1">r1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> holds <br/> ex <font color="Olive" title="b2">t</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">t</font> &amp; <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="limfunc2.html#T4" title="LIMFUNC2:th.4">LIMFUNC2:4</a></i>;<br/>
<a NAME="E4:7"/><i><font color="Green" title="E7">A66</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Olive" title="b1">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font></span>)</span> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>;<br/>

<b>assume </b><a NAME="E1:7_2"/><i><font color="Green" title="E8">A67</font></i>: 
( <font color="Maroon" title="c5">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font></span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:7_2"/><i><font color="Green" title="E9">A68</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="limfunc3.html#T7" title="LIMFUNC3:th.7">LIMFUNC3:7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font>;<br/>
<div><i><font color="Green" title="E10">A69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:7_2_1"/><i><font color="Green" title="E11">A70</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/><a NAME="E2:7_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T10" title="RFUNCT_2:th.10">RFUNCT_2:10</a></i>;<br/><a NAME="E3:7_2_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7_2"><i><font color="Green" title="E8">A67</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:7_2_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g1</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="limfunc1.html#D3" title="LIMFUNC1:def.3">LIMFUNC1:def 3</a></i>;<br/><a NAME="E5:7_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">a</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g1</font> )
 ;<br/><b>hence </b><a NAME="E6:7_2_1"/>
<font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="seqm_3.html#D9" title="SEQM_3:def.9">SEQM_3:def 9</a></i>;<br/><a NAME="E7:7_2_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E8:7_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E3:7_2"><i><font color="Green" title="E9">A68</font></i></a>, <a class="txt" href="l_hospit.html#E1:7_2_1"><i><font color="Green" title="E11">A70</font></i></a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1</span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> $2 );<br/>
<a NAME="E5:7_2"/><i><font color="Green" title="E11">A71</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">c</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:7_2_2"/>
( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E4:7_2"><i><font color="Green" title="E10">A69</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">c</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:7_2_2"/><i><font color="Green" title="E12">A72</font></i>: 
( <font color="Maroon" title="c9">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">c</font> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E2:7"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">c</font>
;<br/>

<a NAME="E4:7_2_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:7_2_2"/><b>then </b><i><font color="Green" title="E13">A73</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_2"><i><font color="Green" title="E8">A67</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E6:7_2_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T7" title="RFUNCT_2:th.7">RFUNCT_2:7</a></i>;<br/>
<a NAME="E7:7_2_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E5:7_2_2"><i><font color="Green" title="E13">A73</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:7_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c9">c</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E3:7_2_2"><i><font color="Green" title="E12">A72</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>;<br/>
<b>hence </b><a NAME="E9:7_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c9">c</font>]
 <b>by </b><i><a class="txt" href="l_hospit.html#E3:7_2_2"><i><font color="Green" title="E12">A72</font></i></a>, <a class="txt" href="l_hospit.html#E5:7_2_2"><i><font color="Green" title="E13">A73</font></i></a>, <a class="ref" href="rfunct_2.html#T22" title="RFUNCT_2:th.22">RFUNCT_2:22</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c8">b</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E7:7_2"/><i><font color="Green" title="E12">A74</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1" title="RCOMP_1:sch.1">RCOMP_1:sch 1</a>(<a class="txt" href="l_hospit.html#E5:7_2"><i><font color="Green" title="E11">A71</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c9">d</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c3">x0</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E9:7_2"/><i><font color="Green" title="E13">A75</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c9">d</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E10:7_2"/><i><font color="Green" title="E14">A76</font></i>: 
<font color="Maroon" title="c9">d</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/>
<i><font color="Green" title="E15">A77</font></i>:  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">d</font> = 
<font color="Maroon" title="c9">d</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 0
<b>by </b><i><a class="ref" href="seq_4.html#T41" title="SEQ_4:th.41">SEQ_4:41</a></i>
<br/>.= 
<font color="Maroon" title="c3">x0</font>
<b>by </b><i><a class="txt" href="l_hospit.html#E9:7_2"><i><font color="Green" title="E13">A75</font></i></a></i>
;<br/>
<a NAME="E12:7_2"/><i><font color="Green" title="E16">A78</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7_2"><i><font color="Green" title="E8">A67</font></i></a>, <a class="ref" href="seq_4.html#T33" title="SEQ_4:th.33">SEQ_4:33</a></i>;<br/>
<div><i><font color="Green" title="E17">A79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><a NAME="E1:7_2_4"/>
<font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <b>by </b><i><a class="txt" href="l_hospit.html#E7:7_2"><i><font color="Green" title="E12">A74</font></i></a></i>;<br/><a NAME="E2:7_2_4"/><b>then </b>
<font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/><a NAME="E3:7_2_4"/><b>then </b>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> )
 ;<br/><b>hence </b><a NAME="E4:7_2_4"/>
( <font color="Maroon" title="c9">d</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E9:7_2"><i><font color="Green" title="E13">A75</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:7_2"/><b>then </b><i><font color="Green" title="E18">A80</font></i>: 
<font color="Maroon" title="c8">b</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E10:7_2"><i><font color="Green" title="E14">A76</font></i></a>, <a class="txt" href="l_hospit.html#E11:7_2"><i><font color="Green" title="E15">A77</font></i></a>, <a class="txt" href="l_hospit.html#E12:7_2"><i><font color="Green" title="E16">A78</font></i></a>, <a class="ref" href="seq_2.html#T33" title="SEQ_2:th.33">SEQ_2:33</a></i>;<br/>
<a NAME="E15:7_2"/><i><font color="Green" title="E19">A81</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E10:7_2"><i><font color="Green" title="E14">A76</font></i></a>, <a class="txt" href="l_hospit.html#E11:7_2"><i><font color="Green" title="E15">A77</font></i></a>, <a class="txt" href="l_hospit.html#E12:7_2"><i><font color="Green" title="E16">A78</font></i></a>, <a class="txt" href="l_hospit.html#E13:7_2"><i><font color="Green" title="E17">A79</font></i></a>, <a class="ref" href="seq_2.html#T34" title="SEQ_2:th.34">SEQ_2:34</a></i>;<br/>
<a NAME="E16:7_2"/><i><font color="Green" title="E20">A82</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_2_5"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:7_2_5"/><i><font color="Green" title="E21">A83</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9" title="RFUNCT_2:th.9">RFUNCT_2:9</a></i>;<br/>
<a NAME="E4:7_2_5"/><i><font color="Green" title="E22">A84</font></i>: 
( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E4:7_2"><i><font color="Green" title="E10">A69</font></i></a></i>;<br/>
<a NAME="E5:7_2_5"/><i><font color="Green" title="E23">A85</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E7:7_2"><i><font color="Green" title="E12">A74</font></i></a>, <a class="txt" href="l_hospit.html#E3:7_2_5"><i><font color="Green" title="E21">A83</font></i></a></i>;<br/>
<a NAME="E6:7_2_5"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g1</font> where <font color="Olive" title="b1">g1</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D2" title="RCOMP_1:def.2">RCOMP_1:def 2</a></i>;<br/>
<a NAME="E7:7_2_5"/><b>then </b>
 ex <font color="Olive" title="b1">g1</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b1">g1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> )
 
;<br/>
<a NAME="E8:7_2_5"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g2</font> where <font color="Olive" title="b1">g2</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">g2</font> </span>}</span> 
 
;<br/>
<a NAME="E9:7_2_5"/><b>then </b><i><font color="Green" title="E24">A86</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="ref" href="limfunc1.html#D3" title="LIMFUNC1:def.3">LIMFUNC1:def 3</a></i>;<br/>
<a NAME="E10:7_2_5"/><i><font color="Green" title="E25">A87</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</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="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T15" title="RCOMP_1:th.15">RCOMP_1:15</a></i>;<br/>
<a NAME="E11:7_2_5"/>
<font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="real_2.html#T174" title="REAL_2:th.174">REAL_2:174</a></i>;<br/>
<a NAME="E12:7_2_5"/><b>then </b>
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g3</font> where <font color="Olive" title="b1">g3</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g3</font> &amp; <font color="Olive" title="b1">g3</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> ) </span>}</span> 
 
;<br/>
<a NAME="E13:7_2_5"/><b>then </b><i><font color="Green" title="E26">A88</font></i>: 
<font color="Maroon" title="c3">x0</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="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E14:7_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">g4</font> where <font color="Olive" title="b1">g4</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( <font color="Maroon" title="c3">x0</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">g4</font> &amp; <font color="Olive" title="b1">g4</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="l_hospit.html#E4:7_2_5"><i><font color="Green" title="E22">A84</font></i></a></i>;<br/>
<a NAME="E15:7_2_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</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="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E16:7_2_5"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></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="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E13:7_2_5"><i><font color="Green" title="E26">A88</font></i></a>, <a class="ref" href="rcomp_1.html#T16" title="RCOMP_1:th.16">RCOMP_1:16</a></i>;<br/>
<a NAME="E17:7_2_5"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</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="c3">x0</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E10:7_2_5"><i><font color="Green" title="E25">A87</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E18:7_2_5"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E19:7_2_5"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font></span>)</span>
 <b>by </b><i><a class="txt" href="l_hospit.html#E5:7_2_5"><i><font color="Green" title="E23">A85</font></i></a>, <a class="txt" href="l_hospit.html#E9:7_2_5"><i><font color="Green" title="E24">A86</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:7_2"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="limfunc1.html#K4" title="LIMFUNC1:func.4">right_open_halfline</a> <font color="Maroon" title="c3">x0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E18:7_2"/><b>then </b><i><font color="Green" title="E21">A89</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c8">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E16:7_2"><i><font color="Green" title="E20">A82</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E19:7_2"/>
 <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font>
 
;<br/>
<a NAME="E20:7_2"/><b>then </b><i><font color="Green" title="E22">A90</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c8">b</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font> )
 
<b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="l_hospit.html#E14:7_2"><i><font color="Green" title="E18">A80</font></i></a>, <a class="txt" href="l_hospit.html#E15:7_2"><i><font color="Green" title="E19">A81</font></i></a>, <a class="txt" href="l_hospit.html#E16:7_2"><i><font color="Green" title="E20">A82</font></i></a>, <a class="ref" href="limfunc2.html#D8" title="LIMFUNC2:def.8">LIMFUNC2:def 8</a></i>;<br/>
<div><i><font color="Green" title="E23">A91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_6"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c10">m</font> = 0;<br/><b>let </b><font color="Maroon" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:7_2_6"/>
<font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">n</font>
 ;<br/><a NAME="E2:7_2_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="l_hospit.html#E7:7_2"><i><font color="Green" title="E12">A74</font></i></a></i>;<br/><b>hence </b><a NAME="E3:7_2_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p4"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">n</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E18:7_2"><i><font color="Green" title="E21">A89</font></i></a>, <a class="ref" href="rfunct_2.html#T21" title="RFUNCT_2:th.21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E22:7_2"/><b>then </b><i><font color="Green" title="E24">A92</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E20:7_2"><i><font color="Green" title="E22">A90</font></i></a>, <a class="ref" href="seq_4.html#T31" title="SEQ_4:th.31">SEQ_4:31</a></i>;<br/>
<a NAME="E23:7_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="seqm_3.html#K1" title="SEQM_3:func.1">^\</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="l_hospit.html#E20:7_2"><i><font color="Green" title="E22">A90</font></i></a>, <a class="txt" href="l_hospit.html#E21:7_2"><i><font color="Green" title="E23">A91</font></i></a>, <a class="ref" href="seq_4.html#T32" title="SEQ_4:th.32">SEQ_4:32</a></i>;<br/>
<b>hence </b><a NAME="E24:7_2"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span> <a href="rfunct_2.html#K1" title="RFUNCT_2:func.1">*</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font> )
 <b>by </b><i><a class="txt" href="l_hospit.html#E22:7_2"><i><font color="Green" title="E24">A92</font></i></a>, <a class="ref" href="seq_4.html#T35" title="SEQ_4:th.35">SEQ_4:35</a>, <a class="ref" href="seq_4.html#T36" title="SEQ_4:th.36">SEQ_4:36</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
<font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font> <a href="limfunc2.html#R4" title="LIMFUNC2:pred.4">is_right_convergent_in</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E3:7"><i><font color="Green" title="E6">A65</font></i></a>, <a class="ref" href="l_hospit.html#T2" target="_self" title="L_HOSPIT:th.2">Th2</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c4">r</font>
;<br/>

<b>thus </b><a NAME="E6:7"/>
<font color="Maroon" title="c4">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="l_hospit.html#E1:7"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:7"/>
 <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <font color="Maroon" title="c2">g</font></span>)</span>,<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc2.html#K2" title="LIMFUNC2:func.2">lim_right</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="rfunct_1.html#K2" title="RFUNCT_1:func.2">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Maroon" title="c3">x0</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">x0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c4">r</font></span>)</span></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="txt" href="l_hospit.html#E3:7"><i><font color="Green" title="E6">A65</font></i></a>, <a class="txt" href="l_hospit.html#E4:7"><i><font color="Green" title="E7">A66</font></i></a>, <a class="ref" href="l_hospit.html#T2" target="_self" title="L_HOSPIT:th.2">Th2</a></i>;<br/>


</div>
