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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><a NAME="E1:133"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R4">are_weakly_separated</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>;<br/>

<a NAME="E2:133"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font> implies ( <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="ref" href="tmap_1.html#T89" target="_self">Th89</a></i>;<br/>

<b>thus </b><a NAME="E4:133"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>5</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></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">E97</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<a NAME="E2:133_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:133_1_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:133_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#R3">are_separated</a> 
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:133"><i><font color="Green">E95</font></i></a>, <a class="ref" href="tsep_1.html#T85">TSEP_1:85</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:133_1_1_1"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E5:133_1_1_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>8</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:133_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="txt" href="tmap_1.html#E5:133_1_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="tmap_1.html#T35" target="_self">Th35</a></i>;<br/>
<a NAME="E7:133_1_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E4:133_1_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="tmap_1.html#T46" target="_self">Th46</a></i>;<br/>
<div><i><font color="Green">E102</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">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><a NAME="E1:133_1_1_1_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:133_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E3:133_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:133_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tmap_1.html#T86" target="_self">Th86</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">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><a NAME="E1:133_1_1_1_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:133_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E3:133_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:133_1_1_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tmap_1.html#T86" target="_self">Th86</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:133_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E8:133_1_1_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:133_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></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">E99</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><i><font color="Green">E100</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">E101</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><div><i><font color="Green">E102</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">c<sub>3</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>4</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/><a NAME="E4:133_1_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E5:133_1_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133_1_1_2_1_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tmap_1.html#T88" target="_self">Th88</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">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_1_2"/><b>then </b><i><font color="Green">E103</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>3</sub></font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/><a NAME="E4:133_1_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E5:133_1_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133_1_1_2_1_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tmap_1.html#T88" target="_self">Th88</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:133_1_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E2:133_1_1_2_1"><i><font color="Green">E102</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"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:133_1_1_2_2"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:133_1_1_2_2"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:133_1_1_2_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:133_1_1_2_2"/><i><font color="Green">E104</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font> or ex <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font> st <br/>(  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font> ) )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133"><i><font color="Green">E95</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <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">E105</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a></i>;<br/><a NAME="E7:133_1_1_2_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is  <a href="tsep_1.html#V1">open</a>  <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="tmap_1.html#T44" target="_self">Th44</a></i>;<br/><div><i><font color="Green">E107</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">E108</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_2_1"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E3:133_1_1_2_2_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a></i>;<br/><a NAME="E4:133_1_1_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></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">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133_1_1_2_2_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="tmap_1.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E6:133_1_1_2_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_2_1"><i><font color="Green">E108</font></i></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">E111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133_1_1_2_2_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>;<br/><div><i><font color="Green">E112</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">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_1_1"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E96</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self">Th90</a></i>;<br/><a NAME="E3:133_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E6:133_1_1_2_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, <a class="ref" href="tmap_1.html#T86" target="_self">Th86</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">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_1_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E96</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self">Th90</a></i>;<br/><a NAME="E3:133_1_1_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E7:133_1_1_2_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="tmap_1.html#T36" target="_self">Th36</a>, <a class="ref" href="tmap_1.html#T86" target="_self">Th86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:133_1_1_2_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:133_1_1_2_2_1"><i><font color="Green">E111</font></i></a>, <a class="txt" href="tmap_1.html#E8:133_1_1_2_2_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self">Th16</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">E108</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> is  not    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></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">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:133_1_1_2_2_2"/><i><font color="Green">E109</font></i>: 
 <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E4:133_1_1_2_2_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E5:133_1_1_2_2"><i><font color="Green">E104</font></i></a></i>;<br/><a NAME="E5:133_1_1_2_2_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="tmap_1.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E6:133_1_1_2_2_2"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="tmap_1.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E7:133_1_1_2_2_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E8:133_1_1_2_2_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self">Th37</a></i>;<br/><a NAME="E9:133_1_1_2_2_2"/>
<font color="Maroon">c<sub>1</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>11</sub></font> =  <a href="pre_topc.html#G1">TopStruct</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font> #) as    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tmap_1.html#T11" target="_self">Th11</a></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">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></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">E115</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font> <a href="tsep_1.html#NR2" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T22" target="_self">Th22</a></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">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E12:133_1_1_2_2_2"><i><font color="Green">E115</font></i></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">E116</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> #)
 <b>by </b><i><a class="txt" href="tmap_1.html#E5:133_1_1_2_2_2"><i><font color="Green">E111</font></i></a>, <a class="txt" href="tmap_1.html#E6:133_1_1_2_2_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>;<br/><div><i><font color="Green">E117</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">E118</font></i>: 
ex <font color="Olive">b<sub>1</sub></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">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_2_2_1"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tsep_1.html#K2">meet</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tsep_1.html#K1">union</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:133_1_1_2_2_2"><i><font color="Green">E113</font></i></a>, <a class="txt" href="tmap_1.html#E8:133_1_1_2_2_2"><i><font color="Green">E114</font></i></a>, <a class="ref" href="tsep_1.html#T35">TSEP_1:35</a></i>;<br/><div><i><font color="Green">E120</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">c<sub>12</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_2_1_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E96</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self">Th90</a></i>;<br/><a NAME="E3:133_1_1_2_2_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E6:133_1_1_2_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_2_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T86" target="_self">Th86</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">c<sub>12</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_2_1_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:133_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133"><i><font color="Green">E96</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="tmap_1.html#T90" target="_self">Th90</a></i>;<br/><a NAME="E3:133_1_1_2_2_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E4:133_1_1_2_2_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="tmap_1.html#E7:133_1_1_2_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="tmap_1.html#E4:133_1_1_2_2_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="tmap_1.html#E1:133_1_1_2_2_2_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="tmap_1.html#T37" target="_self">Th37</a>, <a class="ref" href="tmap_1.html#T86" target="_self">Th86</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:133_1_1_2_2_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_2_2_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="tmap_1.html#E2:133_1_1_2_2_2_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self">Th16</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">c<sub>12</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="tsep_1.html#K2">meet</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span><b> such that </b><a NAME="E1:133_1_1_2_2_2_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K2">meet</a> <font color="Maroon">c<sub>4</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:133_1_1_2_2_2_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E4:133_1_1_2_2_2"><i><font color="Green">E110</font></i></a>, <a class="ref" href="tmap_1.html#T15" target="_self">Th15</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:133_1_1_2_2_2_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E7:133_1_1_2_2_2_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="tmap_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E8:133_1_1_2_2_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#K3" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/><b>hence </b><a NAME="E9:133_1_1_2_2_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1_2_2_2_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="tmap_1.html#E3:133_1_1_2_2_2_2"><i><font color="Green">E119</font></i></a>, <a class="txt" href="tmap_1.html#E5:133_1_1_2_2_2_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="tmap_1.html#E7:133_1_1_2_2_2_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="tmap_1.html#T123" target="_self">Th123</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E18:133_1_1_2_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E15:133_1_1_2_2_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="tmap_1.html#E16:133_1_1_2_2_2"><i><font color="Green">E117</font></i></a>, <a class="ref" href="tmap_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:133_1_1_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E8:133_1_1_2_2"><i><font color="Green">E107</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:133_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E2:133_1_1_2"><i><font color="Green">E100</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:133_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:133_1_1"><i><font color="Green">E98</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:133_1"/>
<font color="Maroon">c<sub>5</sub></font> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T49" target="_self">Th49</a></i>;<br/>


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


</div>
