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<b>let </b><font color="Maroon" title="c6">S</font>, <font color="Maroon" title="c7">T</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c8">A</font>, <font color="Maroon" title="c9">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c7">T</font>;<br/><b>let </b><font color="Maroon" title="c10">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c7">T</font>;<br/>





<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 ;<br/>

<b>given </b><font color="Maroon" title="c11">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">B</font></span>)</span><b> such that </b><a NAME="E2:12"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c7">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">B</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a><br/>

<b>set </b><font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">A</font>;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E21">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E5:12"/><b>then </b>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c11">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T178" title="RELAT_1:th.178">RELAT_1:178</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">Y</font> = <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">B</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E2:12"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">Y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a><br/>

<b>thus </b><a NAME="E7:12"/>
<font color="Maroon" title="c13">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">A</font>
 ;<br/>

<a NAME="E8:12"/>
<font color="Maroon" title="c11">X</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E3:12"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="connsp_1.html#D5" title="CONNSP_1:def.5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E9:12"/><b>then </b>
<font color="Maroon" title="c8">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E2:12"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<a NAME="E10:12"/><b>then </b>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:12"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="tops_2.html#T76" title="TOPS_2:th.76">TOPS_2:76</a></i>;<br/>
<b>hence </b><a NAME="E11:12"/>
<font color="Maroon" title="c13">Y</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D5" title="CONNSP_1:def.5">CONNSP_1:def 5</a><br/>

<b>let </b><font color="Maroon" title="c14">Z</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">B</font></span>)</span></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E12:12"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c14">Z</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E13:12"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c13">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">Z</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c15">Z1</font> = <font color="Maroon" title="c14">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">S</font> <b>by </b><i><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E15:12"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c15">Z1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:12"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<a NAME="E16:12"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">S</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">B</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E17:12"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<i><font color="Green" title="E26">A9</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c7">T</font>
<b>by </b><i><a class="txt" href="topalg_5.html#E1:12"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">T</font>

;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">R</font> = <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">T</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">B</font></span>)</span> <b>by </b><i><a class="txt" href="topalg_5.html#E4:12"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="topalg_5.html#E17:12"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<a NAME="E20:12"/>
<font color="Maroon" title="c10">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:12"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E21:12"/><b>then </b>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">Z1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E15:12"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="tops_2.html#T75" title="TOPS_2:th.75">TOPS_2:75</a></i>;<br/>
<a NAME="E22:12"/><b>then </b><i><font color="Green" title="E27">A10</font></i>: 
<font color="Maroon" title="c16">R</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<a NAME="E23:12"/><i><font color="Green" title="E28">A11</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E1:12"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E24:12"/><i><font color="Green" title="E29">A12</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">Y</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E18:12"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="topalg_5.html#E2:12"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="funct_1.html#T147" title="FUNCT_1:th.147">FUNCT_1:147</a></i>;<br/>
<a NAME="E25:12"/>
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c13">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">Z</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E13:12"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E26:12"/><b>then </b><i><font color="Green" title="E30">A13</font></i>: 
<font color="Maroon" title="c11">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">R</font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E24:12"><i><font color="Green" title="E29">A12</font></i></a>, <a class="txt" href="topalg_5.html#E3:12"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="topalg_5.html#E22:12"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="connsp_1.html#D5" title="CONNSP_1:def.5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E27:12"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">S</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E28:12"/><b>then </b>
<font color="Maroon" title="c15">Z1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f</font>
 
;<br/>
<b>hence </b><a NAME="E29:12"/>
<font color="Maroon" title="c13">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Z</font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E2:12"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="topalg_5.html#E26:12"><i><font color="Green" title="E30">A13</font></i></a>, <a class="txt" href="topalg_5.html#E23:12"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="funct_1.html#T164" title="FUNCT_1:th.164">FUNCT_1:164</a></i>;<br/>


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