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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon">J</font> =  <a href="compts_1.html#K1">1TopSp</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E1:7"/><i><font color="Green">E44</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T8">PCOMPS_1:8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">0</span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">Y</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="tdlat_3.html#T17">TDLAT_3:17</a>, <a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a></i>;<br/>
<a NAME="E3:7"/><i><font color="Green">E45</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
;<br/>
<a NAME="E4:7"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T17">ZFMISC_1:17</a></i>;<br/>
<b>thus </b><a NAME="E5:7"/>
( <font color="Maroon">T</font> is <a href="connsp_1.html#V1">connected</a> implies for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>  holds <br/>(  not <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> or  not <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">T</font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/>

<b>given </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span><b> such that </b><a NAME="E2:7_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E3:7_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">B</font> = <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E4:7_1"/><i><font color="Green">E65</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:7_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<a NAME="E6:7_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E7:7_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> &amp; <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T174">RELAT_1:174</a></i>;<br/>
<a NAME="E8:7_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T141">FUNCT_1:141</a></i>;<br/>
<b>hence </b><a NAME="E9:7_1"/>
contradiction
 <b>by </b><i>, , , , , <a class="ref" href="connsp_1.html#T12">CONNSP_1:12</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E6:7"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>  holds <br/>(  not <font color="Olive">f</font> is <a href="pre_topc.html#V5">continuous</a> or  not <font color="Olive">f</font> is <a href="funct_2.html#V2">onto</a> )
 ;<br/>

<b>assume </b><a NAME="E7:7"/>
not <font color="Maroon">T</font> is <a href="connsp_1.html#V1">connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E9:7"/><i><font color="Green">E69</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E10:7"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E11:7"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E12:7"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E13:7"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E14:7"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T12">CONNSP_1:12</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>;<br/>
<b>reconsider </b><font color="Maroon">j0</font> = 0, <font color="Maroon">j1</font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> = <font color="Maroon">j0</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> = <font color="Maroon">j1</font>;<br/>
<a NAME="E16:7"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> ) )
 
;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span><b> such that </b><br/><a NAME="E18:7"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S5">FUNCT_2:sch 5</a>();<br/></i>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span> ;<br/>
<a NAME="E20:7"/><i><font color="Green">E79</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E21:7"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:7_2_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:7_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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>hence </b><a NAME="E4:7_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E2:7_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:7_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:7_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:7"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_3_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:7_3_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:7_3_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:7_3_1"/><b>then </b>
 not <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E5:7_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E2:7_3"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E3:7_3"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E4:7_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:7_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:7_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:7"/><i><font color="Green">E86</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:7_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="E2:7_4"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_4_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_4_1_1"><b>suppose </b></a><a NAME="E1:7_4_1_1"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_4_1_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:7_4_1_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E5:7_4_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</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>hence </b><a NAME="E6:7_4_1_1"/>
<font color="Maroon">y</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_4_1_2"><b>suppose </b></a><a NAME="E1:7_4_1_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_4_1_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:7_4_1_2"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E5:7_4_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:7_4_1_2"/>
<font color="Maroon">y</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E24:7"/><b>then </b><i><font color="Green">E95</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E25:7"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T171">RELAT_1:171</a></i>;<br/>
<a NAME="E26:7"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p3"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>;<br/>
<a NAME="E27:7"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="compts_1.html#K1">1TopSp</a> <span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span>)</span>  st <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, , , , , , , <a class="ref" href="zfmisc_1.html#T42">ZFMISC_1:42</a></i>;<br/>
<a NAME="E28:7"/><b>then </b>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E29:7"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div>
