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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">xa</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">xb</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> ;<br/>







<b>assume </b><a NAME="E1:4"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">X</font> is <a href="connsp_1.html#V1">connected</a> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1"/><i><font color="Green">E50</font></i>: 
( not <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; not <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><a NAME="E2:4_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:4_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">d</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">rc</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  holds not <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">rc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/><a NAME="E2:4_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/><b>reconsider </b><font color="Maroon">C</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/><div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:4_1_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_1_1"/><i><font color="Green">E65</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">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>thus </b><a NAME="E4:4_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><a NAME="E5:4_1_1"/><i><font color="Green">E66</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/><a NAME="E6:4_1_1"/><i><font color="Green">E67</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">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:4_1_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E8:4_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E9:4_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
not <font color="Maroon">C</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E10:4_1_1"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T28">CONNSP_1:28</a></i>;<br/><b>hence </b><a NAME="E11:4_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:4_1"/>
 ex <font color="Olive">xc</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">xc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:4"/>
 ex <font color="Olive">xc</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> st <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">xc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 <b>by </b><i/>;<br/>


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