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<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:18"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E3:18"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a><br/>

<b>thus </b><a NAME="E4:18"/>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="txt" href="jordan24.html#E1:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan24.html#E2:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jordan24.html#T10" target="_self">Th10</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a><br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E5:18"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<a NAME="E6:18"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan24.html#E1:18"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:18"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T147">FUNCT_1:147</a>, <a class="ref" href="funct_1.html#T164">FUNCT_1:164</a></i>;<br/>
<b>assume </b><a NAME="E8:18"/>
( <font color="Maroon">c<sub>9</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<a NAME="E9:18"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan24.html#E1:18"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan24.html#E7:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="relat_1.html#T178">RELAT_1:178</a>, <a class="ref" href="tops_2.html#T76">TOPS_2:76</a></i>;<br/>
<b>hence </b><a NAME="E10:18"/>
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan24.html#E7:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan24.html#E3:18"><i><font color="Green">E15</font></i></a></i>;<br/>


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