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<b>let </b><font color="Maroon" title="c1">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> iff  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Olive" title="b1">A</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> iff  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span> )</span><br/>



<a NAME="E1:111"/><i><font color="Green" title="E78">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span> <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>thus </b><a NAME="E2:111"/>
( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span> implies <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:111_1"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span></span><br/>

<a NAME="E2:111_1"/><b>then </b><i><font color="Green" title="E79">A2</font></i>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T106" target="_self" title="TMAP_1:th.106">Th106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">f</font> =  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E4:111_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E5:111_1"/><b>then </b>
<font color="Maroon" title="c3">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T94" target="_self" title="TMAP_1:th.94">Th94</a></i>;<br/>
<b>hence </b><a NAME="E6:111_1"/>
 <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:111_1"><i><font color="Green" title="E79">A2</font></i></a>, <a class="ref" href="tmap_1.html#T56" target="_self" title="TMAP_1:th.56">Th56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:111"/>
(  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span> implies <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:111_2"/><i><font color="Green" title="E79">A3</font></i>: 
 <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K6" title="TMAP_1:func.6">modified_with_respect_to</a> <font color="Maroon" title="c2">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a></span><br/>

<b>set </b><font color="Maroon" title="c3">B</font> = <span class="p1">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font>;<br/>
<a NAME="E2:111_2"/><i><font color="Green" title="E80">A4</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></i>;<br/>
<a NAME="E3:111_2"/><i><font color="Green" title="E81">A5</font></i>: 
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T105" target="_self" title="TMAP_1:th.105">Th105</a>, <a class="ref" href="funct_1.html#T162" title="FUNCT_1:th.162">FUNCT_1:162</a></i>;<br/>
<a NAME="E4:111_2"/>
<span class="p1">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">A</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="#E2:111_2"><i><font color="Green" title="E80">A4</font></i></a>, <a class="ref" href="funct_2.html#T171" title="FUNCT_2:th.171">FUNCT_2:171</a></i>;<br/>
<b>hence </b><a NAME="E5:111_2"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E1:111"><i><font color="Green" title="E78">A1</font></i></a>, <a class="txt" href="#E1:111_2"><i><font color="Green" title="E79">A3</font></i></a>, <a class="txt" href="#E3:111_2"><i><font color="Green" title="E81">A5</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
