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<div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</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>7</sub></font> be  <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>,<span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>







<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E1:13"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font> st <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> holds <br/>ex <font color="Olive">b<sub>3</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>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">b<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</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>10</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13_1"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E2:13_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:13_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:13_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E6:13_1"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E7:13_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E8:13_1"/><i><font color="Green">E16</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E9:13_1"/><i><font color="Green">E17</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
<b>by </b><i><a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="topalg_4.html#E7:13_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<a NAME="E11:13_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_4.html#E1:13_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topalg_4.html#E6:13_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="topalg_4.html#E7:13_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="topalg_4.html#E8:13_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:13_1"/>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="topalg_4.html#E2:13_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="borsuk_1.html#T46">BORSUK_1:46</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</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="E14:13_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E15:13_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E16:13_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="topalg_4.html#E11:13_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>13</sub></font>
;<br/>

<b>thus </b><a NAME="E17:13_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="topalg_4.html#E14:13_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topalg_4.html#E15:13_1"><i><font color="Green">E20</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E18:13_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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="E20:13_1"/><i><font color="Green">E22</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="E21:13_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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/>
<a NAME="E22:13_1"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> 
 
<b>by </b><i><a class="txt" href="topalg_4.html#E9:13_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>;<br/>
<a NAME="E23:13_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_4.html#E9:13_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topalg_4.html#E20:13_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E24:13_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="topalg_4.html#E16:13_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topalg_4.html#E21:13_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="topalg_4.html#E22:13_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:13"/>
 <a href="topalg_4.html#K4" target="_self">pr1</a> <font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>


</div>
