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<div class="add">

<a NAME="E1:29_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>6</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>8</sub></font> st <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</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>6</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_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</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="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>8</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:29_1_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E2:29_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>12</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E3:29_1_1_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E5:29_1_1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E6:29_1_1_1"/><i><font color="Green">E32</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="E7:29_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>9</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>12</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E1:29_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="topalg_5.html#E2:29_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="topalg_5.html#E3:29_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jgraph_2.html#T20">JGRAPH_2:20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="borsuk_1.html#K5">:]</a></span><b> such that </b><br/><a NAME="E9:29_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E10:29_1_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</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>6</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="borsuk_1.html#K6">:]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="topalg_5.html#E6:29_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="borsuk_1.html#T45">BORSUK_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:29_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E13:29_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E5:29_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="topalg_5.html#E9:29_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font><b> such that </b><br/><a NAME="E15:29_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>and </b><br/><a NAME="E16:29_1_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E17:29_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E10:29_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="topalg_5.html#E13:29_1_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/>

<b>thus </b><a NAME="E18:29_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E12:29_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topalg_5.html#E15:29_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>

<b>thus </b><a NAME="E19:29_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="topalg_5.html#E16:29_1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>18</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="E20:29_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E22:29_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E23:29_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> = 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="topalg_5.html#D2" target="_self">Def2</a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K7">]</a></span>

;<br/>
<a NAME="E25:29_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E12:29_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="topalg_5.html#E15:29_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E26:29_1_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <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>16</sub></font>,<font color="Maroon">c<sub>17</sub></font></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E22:29_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E27:29_1_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_5.html#E9:29_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="topalg_5.html#E13:29_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="topalg_5.html#E15:29_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E28:29_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="borsuk_1.html#K7">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>
<b>hence </b><a NAME="E29:29_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="topalg_5.html#E7:29_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="topalg_5.html#E23:29_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="topalg_5.html#E24:29_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_1_1"/>
 <a href="topalg_5.html#K3" target="_self">Prj1</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</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>
