<?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>let </b><font color="Maroon">c<sub>5</sub></font> be   <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/>









<b>assume </b><a NAME="E1:83"/><i><font color="Green">E66</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>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E2:83"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:83"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:83"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<b>thus </b><a NAME="E5:83"/>
( <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>7</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> <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#E1:83"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tmap_1.html#E4:83"><i><font color="Green">E69</font></i></a>, <a class="ref" href="tmap_1.html#T81" target="_self">Th81</a></i>;<br/>

<b>thus </b><a NAME="E6:83"/>
( <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> 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>7</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E70</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> <a href="tmap_1.html#R1" target="_self">is_continuous_at</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/>

<a NAME="E2:83_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:83_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:83_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E3:83_1_1"/>
<span class="p1">(<span class="default"><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></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:83"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tmap_1.html#E4:83"><i><font color="Green">E69</font></i></a>, <a class="txt" href="tmap_1.html#E2:83_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="tmap_1.html#T72" target="_self">Th72</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:83_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:83_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E7:83_1_1"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><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></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:83_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="tmap_1.html#E1:83_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tmap_1.html#T48" target="_self">Th48</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font><b> such that </b><br/><a NAME="E9:83_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E10:83_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E11:83_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E3:83"><i><font color="Green">E68</font></i></a>, <a class="ref" href="connsp_2.html#T8">CONNSP_2:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="tmap_1.html#E2:83"><i><font color="Green">E67</font></i></a>, <a class="txt" href="tmap_1.html#E10:83_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E13:83_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E14:83_1_1"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:83"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tmap_1.html#E9:83_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="tops_2.html#T33">TOPS_2:33</a></i>;<br/>
<a NAME="E15:83_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E5:83_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="txt" href="tmap_1.html#E13:83_1_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<a NAME="E17:83_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>assume </b><a NAME="E1:83_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>

<b>then 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="E3:83_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:83_1_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <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>;<br/>
<a NAME="E5:83_1_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E13:83_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="tmap_1.html#E3:83_1_1_1"><i><font color="Green">E81</font></i></a></i>;<br/>
<a NAME="E6:83_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E10:83_1_1"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E7:83_1_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E1:83"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tmap_1.html#E2:83"><i><font color="Green">E67</font></i></a>, <a class="txt" href="tmap_1.html#E4:83_1_1_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="tmap_1.html#T72" target="_self">Th72</a></i>;<br/>
<a NAME="E8:83_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E9:83_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E3:83_1_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="tmap_1.html#E7:83_1_1_1"><i><font color="Green">E83</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E10:83_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E7:83_1_1"><i><font color="Green">E75</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E18:83_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:83"><i><font color="Green">E66</font></i></a>, <a class="txt" href="tmap_1.html#E4:83"><i><font color="Green">E69</font></i></a>, <a class="txt" href="tmap_1.html#E6:83_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="tmap_1.html#E9:83_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="tmap_1.html#E11:83_1_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="tmap_1.html#E13:83_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="tmap_1.html#E15:83_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="subset_1.html#T7">SUBSET_1:7</a>, <a class="ref" href="tsep_1.html#T9">TSEP_1:9</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:83_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>7</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48" target="_self">Th48</a></i>;<br/>


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


</div>
