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<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>





<b>assume </b><a NAME="E1:13"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 ;<br/>

<b>given </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span><b> such that </b><a NAME="E2:13"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>7</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a><br/>

<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E4:13"/><i><font color="Green">E21</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pre_topc.html#K3">|</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>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E5:13"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T178">RELAT_1:178</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E2:13"><i><font color="Green">E19</font></i></a>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a><br/>

<b>thus </b><a NAME="E7:13"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E8:13"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:13"><i><font color="Green">E20</font></i></a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E9:13"/><b>then </b>
<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="topalg_5.html#E2:13"><i><font color="Green">E19</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</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="topalg_5.html#E1:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_2.html#T76">TOPS_2:76</a></i>;<br/>
<b>hence </b><a NAME="E11:13"/>
<font color="Maroon">c<sub>13</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24">CONNSP_1:24</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>14</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E12:13"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>14</sub></font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E13:13"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E15:13"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>15</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:13"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E16:13"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E17:13"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<i><font color="Green">E26</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font> = 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>

;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E4:13"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_5.html#E17:13"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E20:13"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E21:13"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>15</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E15:13"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tops_2.html#T75">TOPS_2:75</a></i>;<br/>
<a NAME="E22:13"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E23:13"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:13"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E24:13"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E18:13"><i><font color="Green">E26</font></i></a>, <a class="txt" href="topalg_5.html#E2:13"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T147">FUNCT_1:147</a></i>;<br/>
<a NAME="E25:13"/>
<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:13"><i><font color="Green">E23</font></i></a>, <a class="ref" href="relat_1.html#T156">RELAT_1:156</a></i>;<br/>
<a NAME="E26:13"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E24:13"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topalg_5.html#E3:13"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topalg_5.html#E22:13"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E27:13"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E28:13"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 
;<br/>
<b>hence </b><a NAME="E29:13"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:13"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_5.html#E26:13"><i><font color="Green">E30</font></i></a>, <a class="txt" href="topalg_5.html#E23:13"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#T164">FUNCT_1:164</a></i>;<br/>


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