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

<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">S</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>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E21">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</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="c2">S</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E22">A2</font></i>: 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a>
 ;<br/>

<a NAME="E4:46"/><i><font color="Green" title="E24">A</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c2">S</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>  &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</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>  )
 
;<br/>
<b>let </b><font color="Maroon" title="c3">X</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D5" target="_self" title="WAYBEL18:def.5">WAYBEL18:def 5</a><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">S</font>;<br/>



<b>assume </b><a NAME="E5:46"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">f'</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="waybel18.html#E1:46"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<a NAME="E7:46"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c5">f'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">P'</font> = <font color="Maroon" title="c6">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E1:46"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<a NAME="E3:46_1"/>
<font color="Maroon" title="c7">P'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:46"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="waybel18.html#E1:46_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:46_1"/><b>then </b>
<font color="Maroon" title="c7">P'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1"/>
<font color="Maroon" title="c5">f'</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c6">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:46"><i><font color="Green" title="E24">A</font></i></a>, <a class="txt" href="waybel18.html#E5:46"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:46"/><b>then </b><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c5">f'</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:46"><i><font color="Green" title="E24">A</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<b>let </b><font color="Maroon" title="c6">Y</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>assume </b><a NAME="E9:46"/>
<font color="Maroon" title="c3">X</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c6">Y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E11:46"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c7">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>and </b><br/><a NAME="E12:46"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c7">h</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f'</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:46"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="waybel18.html#E8:46"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="waybel18.html#D5" target="_self" title="WAYBEL18:def.5">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">Y</font>,<font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel18.html#E1:46"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">g</font>
;<br/>

<a NAME="E14:46"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c8">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font>;<br/>

<b>assume </b><a NAME="E1:46_2"/><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c9">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c10">P'</font> = <font color="Maroon" title="c9">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="waybel18.html#E1:46"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<a NAME="E3:46_2"/>
<font color="Maroon" title="c10">P'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:46"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="waybel18.html#E1:46_2"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E4:46_2"/><b>then </b>
<font color="Maroon" title="c10">P'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:46_2"/>
<font color="Maroon" title="c8">g</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c9">P</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:46"><i><font color="Green" title="E24">A</font></i></a>, <a class="txt" href="waybel18.html#E11:46"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:46"/>
<font color="Maroon" title="c8">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:46"><i><font color="Green" title="E24">A</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>

<b>thus </b><a NAME="E16:46"/>
<font color="Maroon" title="c8">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E12:46"><i><font color="Green" title="E28">A8</font></i></a></i>;<br/>


</div>
