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<b>let </b><font color="Maroon">c<sub>7</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>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>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="borsuk_6.html#R1">are_connected</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>11</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="E2:53"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:53"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E5:53"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="jordan.html#E4:53"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green">E37</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:53"/><i><font color="Green">E38</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="jordan5b.html#T4">JORDAN5B:4</a></i>;<br/>
<a NAME="E9:53"/><b>then </b>
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:53"><i><font color="Green">E34</font></i></a>, <a class="ref" href="borsuk_2.html#D6">BORSUK_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E10:53"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="borsuk_2.html#K2">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan.html#E5:53"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan.html#E7:53"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan.html#E8:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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