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<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>, <font color="Maroon">T</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">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>, <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>, <font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">Q</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><b>let </b><font color="Maroon">P1</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">Q1</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:35"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:35"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="borsuk_6.html#R1">are_connected</a> 
 <b>and </b><br/><a NAME="E3:35"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">P1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:35"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">Q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E5:35"/>
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<a NAME="E1:35_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:35_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c</font> <a href="borsuk_6.html#R1">are_connected</a> 
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_1"/>
( <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> 2 or <font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_1"><b>suppose </b></a><a NAME="E1:35_1_1_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> 2
 ;<br/><div class="add"><a NAME="E2:35_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font> is   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T6">BORSUK_6:6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_1_2"><b>suppose </b></a><a NAME="E1:35_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="xcmplx_0.html#K7">/</a> 2
 ;<br/><div class="add"><a NAME="E2:35_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 is   <a href="pre_topc.html#NM2">Point</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T7">BORSUK_6:7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">Q1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="borsuk_2.html#D5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E6:35"/>
<font color="Maroon">P1</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="borsuk_2.html#K1">+</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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