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

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">thus </span><a NAME="E1:110"/>
( <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="110_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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="E2:110_1"/><span class="lab"><font color="Green" title="E77">A1</font></span>: 
<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/>
<span class="kw">assume </span><a NAME="E3:110_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E4:110_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T95" target="_self" title="TMAP_1:th.95">Th106</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:110_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1"><span class="lab"><font color="Green" title="E77">A1</font></span></a>, <a class="ref" href="tmap_1.html#T51" target="_self" title="TMAP_1:th.51">Th56</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E2:110"/><span class="lab"><font color="Green" title="E77">A2</font></span>: 
 <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/>
<span class="kw">thus </span><a NAME="E3:110"/>
(  <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">assume </span><a NAME="E1:110_2"/><span class="lab"><font color="Green" title="E78">A3</font></span>: 
 <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E2:110_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#K7" title="RELSET_1:func.7">.:</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T92" title="FUNCT_1:th.92">FUNCT_1:92</a></span>;<br/>
<a NAME="E3:110_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A4</font></span>: 
<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#K8" title="RELSET_1:func.8">"</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#K7" title="RELSET_1:func.7">.:</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T94" title="FUNCT_2:th.94">FUNCT_2:94</a></span>;<br/>
<a NAME="E4:110_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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T94" target="_self" title="TMAP_1:th.94">Th105</a>, <a class="ref" href="funct_1.html#T92" title="FUNCT_1:th.92">FUNCT_1:92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:110_2"/>
<font color="Maroon" title="c2">A</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110"><span class="lab"><font color="Green" title="E77">A2</font></span></a>, <a class="txt" href="#E1:110_2"><span class="lab"><font color="Green" title="E78">A3</font></span></a>, <a class="txt" href="#E3:110_2"><span class="lab"><font color="Green" title="E79">A4</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
