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

<b>let </b><font color="Maroon">X</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">Y</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">X1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">X2</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:133"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 ;<br/>

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">Y</font>;<br/>

<a NAME="E2:133"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 
<b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/>
<b>hence </b><a NAME="E3:133"/>
( <font color="Maroon">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">Y</font> implies ( <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Maroon">Y</font> &amp; <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Maroon">Y</font> ) )
 <b>by </b><i><a class="txt" href="tmap_1.html#E12"><i><font color="Green">Lemma69</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:133"/>
( <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Maroon">Y</font> &amp; <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Maroon">Y</font> implies <font color="Maroon">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:133_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X1</font>,<font color="Maroon">Y</font> &amp; <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X2</font>,<font color="Maroon">Y</font> )
 ;<br/>

<a NAME="E2:133_1"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> holds  <font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">X0</font> = <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>;<br/>
<a NAME="E1:133_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> implies <font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:133_1_1_1"/>
<font color="Maroon">X1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font>
 ;<br/>

<a NAME="E2:133_1_1_1"/><b>then </b>
<font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="tsep_1.html#R3">are_separated</a> 
 
<b>by </b><i>, <a class="ref" href="tsep_1.html#T85">TSEP_1:85</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:133_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> &amp; <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y2</font> )
 <b>and </b><br/><a NAME="E5:133_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">Y1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">Y2</font> or <font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">Y2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T83">TSEP_1:83</a></i>;<br/>
<a NAME="E6:133_1_1_1"/>
( <font color="Maroon">Y1</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">Y2</font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, <a class="ref" href="tmap_1.html#T22" target="_self">Th22</a>, </i>;<br/>
<a NAME="E7:133_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">X1</font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 
<b>by </b><i>, <a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E35</font></i></a></i>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X1</font><b> such that </b><a NAME="E1:133_1_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:133_1_1_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="txt" href="tmap_1.html#E5:133_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E3:133_1_1_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X2</font><b> such that </b><a NAME="E1:133_1_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:133_1_1_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="txt" href="tmap_1.html#E5:133_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>hence </b><a NAME="E3:133_1_1_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:133_1_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:133_1_1"/>
( <font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font> implies <font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:133_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X2</font>
 ;<br/>

<div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> or <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/><div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_1_1"/>
<font color="Maroon">X1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font>
 ;<br/><a NAME="E2:133_1_1_2_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X2</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X2</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X2</font> ;<br/><a NAME="E4:133_1_1_2_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E5:133_1_1_2_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_1_2"/>
<font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font>
 ;<br/><a NAME="E2:133_1_1_2_1_2"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X1</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T23">TSEP_1:23</a></i>;<br/><b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X1</font> ;<br/><a NAME="E4:133_1_1_2_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E5:133_1_1_2_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:133_1_1_2_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">X1</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> )
 ;<br/><b>then consider </b><font color="Maroon">Y1</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>, <font color="Maroon">Y2</font> being  non <a href="struct_0.html#V3">empty</a> <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:133_1_1_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font>
 <b>and </b><br/><a NAME="E4:133_1_1_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X2</font>
 <b>and </b><br/><a NAME="E5:133_1_1_2_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font> or  ex <font color="Olive">Z</font> being non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> st <br/>(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Olive">Z</font> &amp; <font color="Olive">Z</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font> ) )
 <b>by </b><i>, , <a class="ref" href="tsep_1.html#T96">TSEP_1:96</a></i>;<br/><a NAME="E6:133_1_1_2_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> implies <font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 <b>by </b><i/>;<br/><a NAME="E7:133_1_1_2_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> implies <font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> )
 <b>by </b><i/>;<br/><div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font>
 ;<br/><a NAME="E2:133_1_1_2_2_1"/><b>then </b><i><font color="Green">E176</font></i>: 
<font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E3:133_1_1_2_2_1"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E29</font></i></a></i>;<br/><a NAME="E4:133_1_1_2_2_1"/>
<font color="Maroon">Y1</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T22">TSEP_1:22</a></i>;<br/><a NAME="E5:133_1_1_2_2_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E6:133_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tsep_1.html#T31">TSEP_1:31</a></i>;<br/><a NAME="E7:133_1_1_2_2_1"/><b>then </b><i><font color="Green">E178</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>;<br/><div><i><font color="Green">E179</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_1_1"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:133_1_1_2_2_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E3:133_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T31" target="_self">Th31</a>, <a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T39" target="_self">Th39</a>, <a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E29</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_1_2"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:133_1_1_2_2_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, </i>;<br/><a NAME="E3:133_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T39" target="_self">Th39</a>, <a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E29</font></i></a>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:133_1_1_2_2_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T38" target="_self">Th38</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_2_2"/><i><font color="Green">E182</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font>
 ;<br/><b>then consider </b><font color="Maroon">Z</font> being  non <a href="struct_0.html#V3">empty</a> <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:133_1_1_2_2_2"/><i><font color="Green">E183</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:133_1_1_2_2_2"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">Z</font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E5:133_1_1_2_2_2"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, </i>;<br/><a NAME="E6:133_1_1_2_2_2"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">Z</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, </i>;<br/><a NAME="E7:133_1_1_2_2_2"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">Y1</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, </i>;<br/><a NAME="E8:133_1_1_2_2_2"/><i><font color="Green">E190</font></i>: 
<font color="Maroon">Y2</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, </i>;<br/><a NAME="E9:133_1_1_2_2_2"/>
<font color="Maroon">X</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T2">TSEP_1:2</a></i>;<br/><b>then reconsider </b><font color="Maroon">X12</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> #) as    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E11:133_1_1_2_2_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X12</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/><a NAME="E12:133_1_1_2_2_2"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X12</font>
 <b>by </b><i><a class="ref" href="tsep_1.html#T4">TSEP_1:4</a></i>;<br/><a NAME="E13:133_1_1_2_2_2"/><b>then </b>
<font color="Maroon">X12</font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font>
 <b>by </b><i/>;<br/><a NAME="E14:133_1_1_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Z</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T44" target="_self">Th44</a>, <a class="ref" href="tsep_1.html#T31">TSEP_1:31</a></i>;<br/><a NAME="E15:133_1_1_2_2_2"/><b>then </b><i><font color="Green">E192</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T38" target="_self">Th38</a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>;<br/><div><i><font color="Green">E193</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_2_2_1"/><i><font color="Green">E194</font></i>: 
 ex <font color="Olive">x12</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> st <font color="Olive">x12</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:133_1_1_2_2_2_1"/><i><font color="Green">E195</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">Y2</font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tmap_1.html#T39" target="_self">Th39</a>, <a class="ref" href="tmap_1.html#T43" target="_self">Th43</a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>;<br/><div><i><font color="Green">E219</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_2_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_2_1_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:133_1_1_2_2_2_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, </i>;<br/><a NAME="E3:133_1_1_2_2_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y1</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_2_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T31" target="_self">Th31</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_2_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_2_1_2"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:133_1_1_2_2_2_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, </i>;<br/><a NAME="E3:133_1_1_2_2_2_1_2"/><b>then </b>
<font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">Y2</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_2_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a>, <a class="ref" href="tmap_1.html#T24" target="_self">Th24</a>, <a class="ref" href="tmap_1.html#T27" target="_self">Th27</a>, <a class="ref" href="tmap_1.html#T32" target="_self">Th32</a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:133_1_1_2_2_2_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tmap_1.html#T47" target="_self">Th47</a>, <a class="ref" href="tmap_1.html#T48" target="_self">Th48</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1_1_2_2_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x0</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_2_2"/><i><font color="Green">E222</font></i>: 
<font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>consider </b><font color="Maroon">x00</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">X2</font></span>)</span><b> such that </b><br/><a NAME="E3:133_1_1_2_2_2_2"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">x00</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, </i>;<br/><b>consider </b><font color="Maroon">x1</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X1</font><b> such that </b><br/><a NAME="E5:133_1_1_2_2_2_2"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x00</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">x2</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X2</font><b> such that </b><br/><a NAME="E7:133_1_1_2_2_2_2"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x00</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:133_1_1_2_2_2_2"/>
( <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X1</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">g</font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">X2</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x2</font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T20" target="_self">Th20</a>, </i>;<br/><b>hence </b><a NAME="E9:133_1_1_2_2_2_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D2" target="_self">Def2</a>, <a class="ref" href="tmap_1.html#T47" target="_self">Th47</a>, <a class="ref" href="tmap_1.html#T48" target="_self">Th48</a>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:133_1_1_2_2_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T46" target="_self">Th46</a>, , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:133_1_1_2_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T33" target="_self">Th33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:133_1_1_2"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:133_1_1"/>
<font color="Maroon">g</font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T21" target="_self">Th21</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:133_1"/>
<font color="Maroon">g</font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">X1</font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">X2</font></span>)</span>,<font color="Maroon">Y</font>
 <b>by </b><i/>;<br/>


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


</div>
