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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:117"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> implies  <a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:117_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:117_1"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T106" target="_self">Th106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E4:117_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:117_1"><i><font color="Green">E88</font></i></a>, <a class="ref" href="tmap_1.html#T107" target="_self">Th107</a></i>;<br/>
<a NAME="E5:117_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T94" target="_self">Th94</a></i>;<br/>
<b>hence </b><a NAME="E6:117_1"/>
 <a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:117_1"><i><font color="Green">E89</font></i></a>, <a class="ref" href="tmap_1.html#T56" target="_self">Th56</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:117"/>
(  <a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> implies <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:117_2"/><i><font color="Green">E88</font></i>: 
 <a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="tmap_1.html#K6" target="_self">modified_with_respect_to</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<i><font color="Green">E89</font></i>: <span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T162">FUNCT_1:162</a></i>
;<br/>
<a NAME="E3:117_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T105" target="_self">Th105</a></i>;<br/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K7" target="_self">modid</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="tmap_1.html#E2:117_2"><i><font color="Green">E89</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="borsuk_1.html#T4">BORSUK_1:4</a></i>
;<br/>
<b>hence </b><a NAME="E5:117_2"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:117_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="tmap_1.html#E3:117_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div>


</div>
