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<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>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="waybel18.html#V2">injective</a>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="borsuk_3.html#R1">are_homeomorphic</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <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><b> such that </b><br/><a NAME="E4:10"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="waybel25.html#E2:10"><i><font color="Green">E7</font></i></a>, <a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a></i>;<br/>
<a NAME="E5:10"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:10"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel18.html#D5">WAYBEL18:def 5</a><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="E6:10"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<b>let </b><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>assume </b><a NAME="E7:10"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E8:10"/>
<font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:10"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E9:10"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel25.html#E6:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:10"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E12:10"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E1:10"><i><font color="Green">E6</font></i></a>, <a class="txt" href="waybel25.html#E7:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="waybel18.html#D5">WAYBEL18:def 5</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>;<br/>

<b>thus </b><a NAME="E13:10"/>
<font color="Maroon">c<sub>9</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="waybel25.html#E5:10"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel25.html#E11:10"><i><font color="Green">E12</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>

<a NAME="E14:10"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E7:10"><i><font color="Green">E11</font></i></a>, <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E15:10"/><b>then </b>
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>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font>
 
;<br/>
<a NAME="E16:10"/><b>then </b><i><font color="Green">E14</font></i>: 
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>6</sub></font>
 
;<br/>
<a NAME="E17:10"/><i><font color="Green">E15</font></i>: 
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="funct_1.html#V2">one-to-one</a> )
 
<b>by </b><i><a class="txt" href="waybel25.html#E4:10"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tops_2.html#T72">TOPS_2:72</a></i>;<br/>
<i><font color="Green">E16</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> = 
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/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K3">/\</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="txt" href="waybel25.html#E16:10"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K8">/\</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/>
<a NAME="E19:10"/>
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>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:10_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E3:10_2"/><b>then </b>
<font color="Maroon">c<sub>10</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>by </b><i><a class="txt" href="waybel25.html#E16:10"><i><font color="Green">E14</font></i></a></i>;<br/>
<a NAME="E4:10_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:10_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="waybel25.html#E1:10_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:10_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<a NAME="E7:10_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E4:10_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E12:10"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel25.html#E1:10_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E1:10_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E7:10_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="waybel25.html#E17:10"><i><font color="Green">E15</font></i></a>, <a class="ref" href="tops_2.html#T65">TOPS_2:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="waybel25.html#E5:10_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E20:10"/>
<font color="Maroon">c<sub>9</sub></font> <a href="partfun1.html#K2">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="waybel25.html#E18:10"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div>
