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<b>let </b><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>given </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font><b> such that </b><a NAME="E1:11"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<b>assume </b><a NAME="E2:11"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>5</sub></font> holds <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D3">BORSUK_2:def 3</a><br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>6</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D3">BORSUK_2:def 3</a><br/>

<a NAME="E3:11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:11"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E5:11"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E6:11"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E7:11"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>10</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1">BORSUK_2:def 1</a><br/>

<a NAME="E8:11"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:11"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:11"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="topalg_3.html#E5:11"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>

<a NAME="E10:11"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:11"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E11:11"/><i><font color="Green">E12</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:11"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 0 = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E6:11"><i><font color="Green">E9</font></i></a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="topalg_3.html#E10:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topalg_3.html#E11:11"><i><font color="Green">E12</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="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>6</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="topalg_3.html#E11:11"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>

<b>thus </b><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E7:11"><i><font color="Green">E10</font></i></a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="topalg_3.html#E10:11"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topalg_3.html#E11:11"><i><font color="Green">E12</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="grcat_1.html#K7">id</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="topalg_3.html#E11:11"><i><font color="Green">E12</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/>


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