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<div class="add">

<b>let </b><font color="Maroon" title="c4">S</font>, <font color="Maroon" title="c5">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">S</font>,<font color="Maroon" title="c5">T</font>;<br/><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c4">S</font>;<br/><b>let </b><font color="Maroon" title="c10">P</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>;<br/><b>let </b><font color="Maroon" title="c11">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font>;<br/><b>let </b><font color="Maroon" title="c12">P1</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font>;<br/><b>let </b><font color="Maroon" title="c13">Q1</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font>;<br/>













<b>assume </b><b>that </b><br/><a NAME="E1:36"/><i><font color="Green" title="E14">A1</font></i>: 
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E2:36"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c9">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c12">P1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">P</font>
 <b>and </b><br/><a NAME="E4:36"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c13">Q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">Q</font>
 ;<br/>

<a NAME="E5:36"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c12">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">Q1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">Q</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> ;<br/>

<a NAME="E1:36_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_3.html#E1:36"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="topalg_3.html#T24" target="_self" title="TOPALG_3:th.24">Th24</a></i>;<br/>
<a NAME="E2:36_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">c</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 
<b>by </b><i><a class="txt" href="topalg_3.html#E2:36"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="topalg_3.html#T24" target="_self" title="TOPALG_3:th.24">Th24</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:36_1_1"/>
( <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 or <font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_1"><b>suppose </b></a><a NAME="E1:36_1_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ;<br/><div class="add"><a NAME="E2:36_1_1_1"/><b>then </b><i><font color="Green" title="E21">A8</font></i>: 
2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T6" title="BORSUK_6:th.6">BORSUK_6:6</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c12">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">Q1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> = 
<font color="Maroon" title="c12">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:36_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="topalg_3.html#E2:36_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topalg_3.html#E1:36_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E3:36"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="topalg_3.html#E2:36_1_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:36"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topalg_3.html#E2:36"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="topalg_3.html#E1:36_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">Q</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_1_2"><b>suppose </b></a><a NAME="E1:36_1_1_2"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 ;<br/><div class="add"><a NAME="E2:36_1_1_2"/><b>then </b><i><font color="Green" title="E21">A10</font></i>: 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="borsuk_1.html#K19" title="BORSUK_1:func.19">I[01]</a> 
 <b>by </b><i><a class="ref" href="borsuk_6.html#T7" title="BORSUK_6:th.7">BORSUK_6:7</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c12">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">Q1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> = 
<font color="Maroon" title="c13">Q1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:36_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="topalg_3.html#E2:36_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="topalg_3.html#E1:36_1_1_2"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E4:36"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="topalg_3.html#E2:36_1_1_2"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">Q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topalg_3.html#E1:36"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="topalg_3.html#E2:36"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="topalg_3.html#E1:36_1_1_2"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="borsuk_2.html#D5" title="BORSUK_2:def.5">BORSUK_2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">Q</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">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="E6:36"/>
<font color="Maroon" title="c12">P1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c13">Q1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">P</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c11">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div>
