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

<b>let </b><font color="Maroon" title="c5">T1</font>, <font color="Maroon" title="c6">T2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c7">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">T1</font>,<font color="Maroon" title="c6">T2</font>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</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="c5">T1</font>;<br/>

<b>given </b><font color="Maroon" title="c10">A1</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="c5">T1</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="E3:22"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c10">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">A</font> &amp; <font color="Maroon" title="c10">A1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c5">T1</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/>

<a NAME="E4:22"/><i><font color="Green" title="E19">A3</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">T1</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> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">A2</font> = <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</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">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="jordan24.html#E3:22"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green" title="E20">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">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="c5">T1</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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1"/>
( <font color="Maroon" title="c9">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c9">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1"><b>suppose </b></a><a NAME="E1:22_1_1"/>
<font color="Maroon" title="c9">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:22_1_1"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T149" title="RELAT_1:th.149">RELAT_1:149</a></i>;<br/><a NAME="E3:22_1_1"/><b>then </b>
( <font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c6">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font> )
 <b>by </b><i><a class="txt" href="jordan24.html#E4:22"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E4:22_1_1"/>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="ref" href="jordan1k.html#T6" title="JORDAN1K:th.6">JORDAN1K:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_2"><b>suppose </b></a><a NAME="E1:22_1_2"/><i><font color="Green" title="E21">A5</font></i>: 
not <font color="Maroon" title="c9">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:22_1_2"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="jordan24.html#E6:22"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">S1</font> = <font color="Maroon" title="c5">T1</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c9">B</font>, <font color="Maroon" title="c13">S2</font> = <font color="Maroon" title="c6">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font></span>)</span> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> <b>by </b><i><a class="txt" href="jordan24.html#E4:22"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="jordan24.html#E1:22_1_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c11">A2</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="E4:22_1_2"/>
<font color="Maroon" title="c11">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font>
 ;<br/><b>reconsider </b><font color="Maroon" title="c14">fB</font> = <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c9">B</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c12">S1</font>,<font color="Maroon" title="c13">S2</font> <b>by </b><i><a class="ref" href="jordan24.html#T12" target="_self" title="JORDAN24:th.12">Th12</a></i>;<br/><a NAME="E6:22_1_2"/>
( <font color="Maroon" title="c14">fB</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">A2</font> &amp; <font color="Maroon" title="c14">fB</font> is <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> )
 <b>by </b><i><a class="txt" href="jordan24.html#E1:22"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="jordan24.html#E3:22"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="jordan24.html#E4:22"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="jordan24.html#T14" target="_self" title="JORDAN24:th.14">Th14</a>, <a class="ref" href="relat_1.html#T162" title="RELAT_1:th.162">RELAT_1:162</a></i>;<br/><b>hence </b><a NAME="E7:22_1_2"/>
<font color="Maroon" title="c11">A2</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c6">T2</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan24.html#E3:22"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="jordan24.html#T11" target="_self" title="JORDAN24:th.11">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
