<?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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X0</font> being   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> iff  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Olive" title="b1">X0</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">X0</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> iff  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:132"/>
( <font color="Maroon" title="c2">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> implies  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span> implies <font color="Maroon" title="c2">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:132_1"/>
<font color="Maroon" title="c2">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span></span><br/>

<a NAME="E2:132_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A1</font></span>: 
<font color="Maroon" title="c3">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="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></span>;<br/>
<a NAME="E3:132_1"/>
( <font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</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="c3">A</font> &amp;  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D10" target="_self" title="TMAP_1:def.10">Def10</a>, <a class="ref" href="tmap_1.html#D11" target="_self" title="TMAP_1:def.11">Def11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:132_1"/>
 <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:132_1"><span class="lab"><font color="Green" title="E80">A1</font></span></a>, <a class="ref" href="tmap_1.html#T113" target="_self" title="TMAP_1:th.113">Th113</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E3:132"/>
(  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span> implies <font color="Maroon" title="c2">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:132_2"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
 <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</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#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:132_2"/>
( <font color="Maroon" title="c1">X</font> <a href="tmap_1.html#K8" title="TMAP_1:func.8">modified_with_respect_to</a> <font color="Maroon" title="c2">X0</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="c3">A</font> &amp;  <a href="tmap_1.html#K9" title="TMAP_1:func.9">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">X0</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="tmap_1.html#K7" title="TMAP_1:func.7">modid</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c3">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#D10" target="_self" title="TMAP_1:def.10">Def10</a>, <a class="ref" href="tmap_1.html#D11" target="_self" title="TMAP_1:def.11">Def11</a></span>;<br/>
<a NAME="E3:132_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="#E1:132_2"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="ref" href="tmap_1.html#T113" target="_self" title="TMAP_1:th.113">Th113</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:132_2"/>
<font color="Maroon" title="c2">X0</font> is   <a href="tsep_1.html#V1" title="TSEP_1:attr.1">open</a>   <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tsep_1.html#T16" title="TSEP_1:th.16">TSEP_1:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
