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

<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:88"/><i><font color="Green" title="E54">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="pscomp_1.html#V9" title="PSCOMP_1:attr.9">continuous</a>
 <b>and </b><br/><a NAME="E2:88"/><i><font color="Green" title="E55">A2</font></i>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c2">f</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">X0</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D25" target="_self" title="PSCOMP_1:def.25">PSCOMP_1:def 25</a><br/>

<b>assume </b><a NAME="E3:88"/><i><font color="Green" title="E56">A3</font></i>: 
<font color="Maroon" title="c3">X0</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 ;<br/>

<a NAME="E4:88"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4: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/>
<a NAME="E5:88"/><b>then </b><i><font color="Green" title="E57">A4</font></i>: 
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4: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="E6:88"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X0</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c3">X0</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> as  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">without_zero</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="txt" href="pscomp_1.html#E5:88"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="pscomp_1.html#D1" target="_self" title="PSCOMP_1:def.1">Def1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">X'</font> =  <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">X'r</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:88_1_1"/><i><font color="Green" title="E59">A6</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><a NAME="E2:88_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T33" target="_self" title="PSCOMP_1:th.33">Th33</a></i>;<br/><a NAME="E3:88_1_1"/><b>then </b>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">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="c2">f</font> )
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:88_1_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:88_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">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/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:88_1"/><i><font color="Green" title="E59">A7</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><a NAME="E3:88_1"/><b>then </b><i><font color="Green" title="E60">A8</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c7">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="c2">f</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="pscomp_1.html#E2:88_1"><i><font color="Green" title="E59">A7</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c9">seq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:88_1"/><i><font color="Green" title="E61">A9</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">seq</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>and </b><br/><a NAME="E7:88_1"/><i><font color="Green" title="E62">A10</font></i>: 
<font color="Maroon" title="c9">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E8:88_1"/><i><font color="Green" title="E63">A11</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:88_1"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="pscomp_1.html#T39" target="_self" title="PSCOMP_1:th.39">Th39</a></i>;<br/><b>assume </b><a NAME="E9:88_1"/>
not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/><a NAME="E10:88_1"/><b>then </b><i><font color="Green" title="E64">A12</font></i>: 
not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:88_1"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:88_1_2"/><i><font color="Green" title="E65">A13</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_2_1"><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/><b>assume </b><a NAME="E1:88_1_2_1"/>
<font color="Maroon" title="c9">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E2:88_1_2_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c9">seq</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/><b>hence </b><a NAME="E3:88_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:88"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E6:88_1"><i><font color="Green" title="E61">A9</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="E3:88_1_2"/><b>then </b><i><font color="Green" title="E66">A14</font></i>: 
<font color="Maroon" title="c9">seq</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7" title="SEQ_1:th.7">SEQ_1:7</a></i>;<br/><a NAME="E4:88_1_2"/><b>then </b><i><font color="Green" title="E67">A15</font></i>: 
<font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E7:88_1"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="pscomp_1.html#E1:88_1_2"><i><font color="Green" title="E65">A13</font></i></a>, <a class="ref" href="seq_2.html#T35" title="SEQ_2:th.35">SEQ_2:35</a></i>;<br/><a NAME="E5:88_1_2"/><i><font color="Green" title="E68">A16</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c9">seq</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E7:88_1"><i><font color="Green" title="E62">A10</font></i></a>, <a class="txt" href="pscomp_1.html#E1:88_1_2"><i><font color="Green" title="E65">A13</font></i></a>, <a class="txt" href="pscomp_1.html#E3:88_1_2"><i><font color="Green" title="E66">A14</font></i></a>, <a class="ref" href="seq_2.html#T36" title="SEQ_2:th.36">SEQ_2:36</a></i>;<br/><a NAME="E6:88_1_2"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="E1:88_1_2_2"/>
<font color="Maroon" title="c10">y</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> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:88_1_2_2"/><i><font color="Green" title="E69">A17</font></i>: 
( <font color="Maroon" title="c11">n</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="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c11">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">n</font> as    <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>by </b><i><a class="txt" href="pscomp_1.html#E3:88_1_2_2"><i><font color="Green" title="E69">A17</font></i></a></i>;<br/>
<a NAME="E5:88_1_2_2"/>
<font color="Maroon" title="c9">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</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="c9">seq</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E6:88_1_2_2"/><b>then </b>
<font color="Maroon" title="c9">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E6:88_1"><i><font color="Green" title="E61">A9</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="E7:88_1_2_2"/><b>then </b><i><font color="Green" title="E70">A18</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T56" title="XCMPLX_1:th.56">XCMPLX_1:56</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
<b>by </b><i><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></i>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c12">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217" title="XCMPLX_1:th.217">XCMPLX_1:217</a></i>
;<br/>
<b>hence </b><a NAME="E9:88_1_2_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:88_1_2_2"><i><font color="Green" title="E69">A17</font></i></a>, <a class="txt" href="pscomp_1.html#E7:88_1_2_2"><i><font color="Green" title="E70">A18</font></i></a>, <a class="ref" href="pscomp_1.html#T28" target="_self" title="PSCOMP_1:th.28">Th28</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:88_1_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="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X0</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E6:88"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:88_1_2"/><b>then </b><i><font color="Green" title="E69">A19</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:88"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="pscomp_1.html#E4:88_1_2"><i><font color="Green" title="E67">A15</font></i></a>, <a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a></i>;<br/><a NAME="E9:88_1_2"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:88_1_2"><i><font color="Green" title="E65">A13</font></i></a>, <a class="txt" href="pscomp_1.html#E5:88_1_2"><i><font color="Green" title="E68">A16</font></i></a>, <a class="ref" href="xcmplx_1.html#T203" title="XCMPLX_1:th.203">XCMPLX_1:203</a></i>;<br/><a NAME="E10:88_1_2"/><b>then </b>
not  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4: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/><a NAME="E11:88_1_2"/><b>then </b>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:88_1_2"><i><font color="Green" title="E69">A19</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="E12:88_1_2"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font>
 ;<br/><b>hence </b><a NAME="E13:88_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:88_1"><i><font color="Green" title="E63">A11</font></i></a>, <a class="txt" href="pscomp_1.html#E10:88_1"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="pscomp_1.html#E5:88_1_2"><i><font color="Green" title="E68">A16</font></i></a>, <a class="ref" href="xcmplx_1.html#T218" title="XCMPLX_1:th.218">XCMPLX_1:218</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:88_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:88"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E2:88_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="txt" href="pscomp_1.html#E8:88_1"><i><font color="Green" title="E63">A11</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="E9:88"/><b>then </b><i><font color="Green" title="E59">A20</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E10:88"/>
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:88"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="pscomp_1.html#D25" target="_self" title="PSCOMP_1:def.25">Def25</a></i>;<br/>
<a NAME="E11:88"/><b>then </b><i><font color="Green" title="E60">A21</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E9:88"><i><font color="Green" title="E59">A20</font></i></a>, <a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>;<br/>
<a NAME="E12:88"/><i><font color="Green" title="E61">A22</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>;<br/>
<a NAME="E13:88"/><i><font color="Green" title="E62">A23</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" title="PSCOMP_1:func.5">Inv</a> <font color="Maroon" title="c4">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T46" target="_self" title="PSCOMP_1:th.46">Th46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:88_2_1"/><i><font color="Green" title="E63">A24</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">X0</font>
 ;<br/><a NAME="E2:88_2_1"/><b>then </b><i><font color="Green" title="E64">A25</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X0</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:88_2_1_1"/>
not <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 ;<br/><a NAME="E2:88_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:88_2_1"><i><font color="Green" title="E64">A25</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="E3:88_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:88_2_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> 
 <b>by </b><i><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></i>;<br/><b>hence </b><a NAME="E5:88_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:88"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E1:88_2_1"><i><font color="Green" title="E63">A24</font></i></a>, <a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a>, <a class="ref" href="xcmplx_1.html#T203" title="XCMPLX_1:th.203">XCMPLX_1:203</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:88_2_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:88_2_1"><i><font color="Green" title="E63">A24</font></i></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></i>;<br/></div>
<b>end;</b></div><a NAME="E2:88_2"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">X0</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>hence </b><a NAME="E3:88_2"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">X0</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:88"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">Inv</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c3">X0</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E11:88"><i><font color="Green" title="E60">A21</font></i></a>, <a class="txt" href="pscomp_1.html#E12:88"><i><font color="Green" title="E61">A22</font></i></a>, <a class="txt" href="pscomp_1.html#E13:88"><i><font color="Green" title="E62">A23</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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