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

<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:91"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> is <a href="pscomp_1.html#V9" target="_self">continuous</a>
 <b>and </b><br/><a NAME="E2:91"/><i><font color="Green">E31</font></i>: 
not 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

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

<b>assume </b><a NAME="E3:91"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X0</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>

<a NAME="E4:91"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:91"/><b>then </b><i><font color="Green">E34</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E6:91"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X0</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X0</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X0</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span> as  <a href="setfam_1.html#V1">without_zero</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">X'</font> =  <a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">X'r</font> = <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:91_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:91_1_1"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:91"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E3:91_1_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:91_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:91_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E3:91_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">seq</font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E6:91_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E7:91_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">seq</font> is <a href="seq_2.html#V4">convergent</a>
 <b>and </b><br/><a NAME="E8:91_1"/><i><font color="Green">E45</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="txt" href="pscomp_1.html#E6:91"><i><font color="Green">E35</font></i></a></i>;<br/><b>assume </b><a NAME="E9:91_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E10:91_1"/><b>then </b><i><font color="Green">E46</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:91_1_2_1"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:91_1_2_1"/><b>then </b>
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/><b>hence </b><a NAME="E3:91_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E12:91"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E3:91_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#NR1" target="_self">is_not_0</a> 
 <b>by </b><i><a class="ref" href="seq_1.html#T7">SEQ_1:7</a></i>;<br/><a NAME="E4:91_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a>  is <a href="seq_2.html#V4">convergent</a>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T2" target="_self">Th2</a>, , <a class="ref" href="seq_2.html#T35">SEQ_2:35</a></i>;<br/><a NAME="E5:91_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="real_1.html#K2">"</a> 
 <b>by </b><i><a class="ref" href="pscomp_1.html#T2" target="_self">Th2</a>, , , <a class="ref" href="seq_2.html#T36">SEQ_2:36</a></i>;<br/><a NAME="E6:91_1_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:91_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:91_1_2_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E5:91_1_2_2"/>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">seq</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/>
<a NAME="E6:91_1_2_2"/><b>then </b>
<font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:91_1_2_2"/><b>then </b><i><font color="Green">E54</font></i>: 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T56">XCMPLX_1:56</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
;<br/>
<b>hence </b><a NAME="E9:91_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T4" target="_self">Th4</a>, <a class="ref" href="pscomp_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:91_1_2"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D2" target="_self">Def2</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:91_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font>
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="rcomp_1.html#D4">RCOMP_1:def 4</a></i>;<br/><a NAME="E9:91_1_2"/>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a></i>;<br/><a NAME="E10:91_1_2"/><b>then </b>
not  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E11:91_1_2"/><b>then </b>
 <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E12:91_1_2"/><b>then </b>
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <span class="p2">(<span class="default"><font color="Maroon">seq</font> <a href="seq_1.html#K18">"</a> </span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font>
 ;<br/><b>hence </b><a NAME="E13:91_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T218">XCMPLX_1:218</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:91_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E12:91"><i><font color="Green">E59</font></i></a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:91"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E10:91"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K6" target="_self">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:91"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>;<br/>
<a NAME="E12:91"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T168">RELAT_1:168</a></i>;<br/>
<a NAME="E13:91"/><i><font color="Green">E176</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K5" target="_self">Inv</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:91_2_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X0</font>
 ;<br/><a NAME="E2:91_2_1"/><b>then </b><i><font color="Green">E178</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X0</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:91_2_1_1"/>
not <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:91_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default">0</span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:91_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then </b>0 = 
1 <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T217">XCMPLX_1:217</a></i>
;<br/><b>hence </b><a NAME="E5:91_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pscomp_1.html#E12:91"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xcmplx_1.html#T203">XCMPLX_1:203</a>, , <a class="ref" href="funct_2.html#T6">FUNCT_2:6</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:91_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/></div>
<b>end;</b></div><a NAME="E2:91_2"/>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/><a NAME="E3:91_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X0</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/><b>hence </b><a NAME="E4:91_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X0</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:91"/>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K9" target="_self">Inv</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">X0</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , <a class="ref" href="pscomp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
