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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">X0</font>, <font color="Olive" title="b2">X1</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">X0</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Olive" title="b2">X1</font> holds <br/>for <font color="Olive" title="b3">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b2">X1</font> holds  <span class="p1">(<span class="default"><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></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Olive" title="b2">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b3">x1</font></span><br/><b>let </b><font color="Maroon" title="c2">X0</font>, <font color="Maroon" title="c3">X1</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X0</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X1</font> implies for <font color="Olive" title="b1">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font> holds  <span class="p1">(<span class="default"><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></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x1</font> )</span><br/>



<b>assume </b><a NAME="E1:119"/><i><font color="Green" title="E82">A1</font></i>: 
<font color="Maroon" title="c2">X0</font> <a href="tsep_1.html#R1" title="TSEP_1:pred.1">misses</a> <font color="Maroon" title="c3">X1</font>
 ; <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">x1</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font> holds  <span class="p1">(<span class="default"><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></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">x1</font></span><br/>

<b>let </b><font color="Maroon" title="c4">x1</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">X1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><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></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c4">x1</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">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> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<a NAME="E3:119"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="#E1:119"><i><font color="Green" title="E82">A1</font></i></a>, <a class="ref" href="tsep_1.html#D3" title="TSEP_1:def.3">TSEP_1:def 3</a></i>;<br/>
<a NAME="E4:119"/><b>then </b><i><font color="Green" title="E83">A2</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="c5">A</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c4">x1</font>
 
<b>by </b><i><a class="ref" href="tmap_1.html#T109" target="_self" title="TMAP_1:th.109">Th109</a></i>;<br/>
<a NAME="E5:119"/>
<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="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 
<b>by </b><i><a class="ref" href="tmap_1.html#D10" target="_self" title="TMAP_1:def.10">Def10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">f</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="c5">A</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">X1</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> ;<br/>
<a NAME="E7:119"/>
<font color="Maroon" title="c6">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c4">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> implies  ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font> st <br/>( <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font> ) )</span><br/><b>assume </b><a NAME="E1:119_1_1"/><i><font color="Green" title="E84">A3</font></i>: 
( <font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font> st <br/>( <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font> )</span><br/><a NAME="E2:119_1_1"/><b>then </b>
<font color="Maroon" title="c7">W</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 <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>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><a NAME="E3:119_1_1"/><b>then </b><i><font color="Green" title="E85">A4</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> <a href="tmap_1.html#K5" title="TMAP_1:func.5">-extension_of_the_topology_of</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T114" target="_self" title="TMAP_1:th.114">Th114</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">W0</font> = <font color="Maroon" title="c7">W</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <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="c5">A</font></span>)</span> <b>by </b><i><a class="ref" href="tmap_1.html#T114" target="_self" title="TMAP_1:th.114">Th114</a></i>;<br/><a NAME="E5:119_1_1"/><i><font color="Green" title="E86">A5</font></i>: 
<font color="Maroon" title="c8">W0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="#E3:119_1_1"><i><font color="Green" title="E85">A4</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/><b>thus </b><a NAME="E6:119_1_1"/>
 ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font> st <br/>( <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font> )
  <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="119_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c9">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">X1</font><b> such that </b><br/><a NAME="E2:119_1_1_1"/><i><font color="Green" title="E87">A6</font></i>: 
( <font color="Maroon" title="c9">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; <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="c5">A</font></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">W0</font> )
 <b>by </b><i><a class="txt" href="#E4:119"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="#E1:119_1_1"><i><font color="Green" title="E84">A3</font></i></a>, <a class="txt" href="#E5:119_1_1"><i><font color="Green" title="E86">A5</font></i></a>, <a class="ref" href="tmap_1.html#T48" target="_self" title="TMAP_1:th.48">Th48</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">V</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font> )</span><br/>

<b>thus </b><a NAME="E3:119_1_1_1"/>
( <font color="Maroon" title="c9">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">V</font> &amp; <font color="Maroon" title="c6">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font> )
 <b>by </b><i><a class="txt" href="#E2:119_1_1_1"><i><font color="Green" title="E87">A6</font></i></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><b>end;</b></div>
<b>hence </b><a NAME="E2:119_1"/>
<font color="Maroon" title="c6">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c4">x1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48" target="_self" title="TMAP_1:th.48">Th48</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>hence </b><a NAME="E8:119"/>
<span class="p1">(<span class="default"><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></span>)</span> <a href="tmap_1.html#K2" title="TMAP_1:func.2">|</a> <font color="Maroon" title="c3">X1</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c4">x1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D11" target="_self" title="TMAP_1:def.11">Def11</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>
