<?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> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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="E1:132"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="tsep_1.html#K1">union</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</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>5</sub></font>;<br/>





<b>assume </b><a NAME="E2:132"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E3:132"/>
( <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>6</sub></font> implies ( <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" 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>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <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>8</sub></font> ) )
 <b>by </b><i><a class="ref" href="tmap_1.html#T64" target="_self">Th64</a></i>;<br/>

<b>thus </b><a NAME="E4:132"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" 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>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <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>8</sub></font> implies <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>6</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:132_1"/><i><font color="Green">E96</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" 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>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <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>8</sub></font> )
 ;<br/>

<a NAME="E2:132_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>6</sub></font> st <font color="Maroon">c<sub>3</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="132_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E1:132_1_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="tmap_1.html#E2:132"><i><font color="Green">E95</font></i></a>, <a class="ref" href="tmap_1.html#T58" target="_self">Th58</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E3:132_1_1"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</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#E1:132_1"><i><font color="Green">E96</font></i></a>, <a class="ref" href="tmap_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E5:132_1_1"/><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tmap_1.html#K2" target="_self">|</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>11</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#E1:132_1"><i><font color="Green">E96</font></i></a>, <a class="txt" href="tmap_1.html#E1:132_1_1"><i><font color="Green">E97</font></i></a>, <a class="ref" href="tmap_1.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E7:132_1_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="tmap_1.html#E1:132"><i><font color="Green">E94</font></i></a>, <a class="txt" href="tmap_1.html#E2:132"><i><font color="Green">E95</font></i></a>, <a class="ref" href="tmap_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<a NAME="E8:132_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:132_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</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>1</sub></font> <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:132_1_1"/>
( <font color="Maroon">c<sub>3</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> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>14</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#E3:132_1_1"><i><font color="Green">E98</font></i></a>, <a class="txt" href="tmap_1.html#E5:132_1_1"><i><font color="Green">E99</font></i></a>, <a class="ref" href="tmap_1.html#T61" target="_self">Th61</a></i>;<br/>
<a NAME="E13:132_1_1"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="ref" href="tmap_1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E14:132_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E15:132_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="tmap_1.html#E7:132_1_1"><i><font color="Green">E100</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E16:132_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:132_1"/>
<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>6</sub></font>
 <b>by </b><i><a class="ref" href="tmap_1.html#D2" target="_self">Def2</a></i>;<br/>


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


</div>
