<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">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="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">P</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/><b>let </b><font color="Maroon" title="c5">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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>









<b>assume </b><a NAME="E1:58"/><i><font color="Green" title="E45">A1</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a>  )
 ;<br/>

<b>set </b><font color="Maroon" title="c6">Q</font> =  <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">f</font>;<br/>
<a NAME="E2:58"/><i><font color="Green" title="E46">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; 1 is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">X</font> = <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">ID</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of  <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">0[01]</a> , <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">1[01]</a>  <b>by </b><i><a class="ref" href="borsuk_6.html#T12" target="_self" title="BORSUK_6:th.12">Th12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">G1</font> =  <a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c8">ID</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">G2</font> =  <a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as  <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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T40" title="YELLOW12:th.40">YELLOW12:40</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">f1</font> = <span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c8">ID</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">G2</font> as  <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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E6:58"/><i><font color="Green" title="E47">A3</font></i>: 
for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c11">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">s</font>, <font color="Maroon" title="c13">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:58_1"/><i><font color="Green" title="E48">A4</font></i>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="jordan5b.html#T4" title="JORDAN5B:th.4">JORDAN5B:4</a></i>;<br/>
<a NAME="E2:58_1"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:58_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">G2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b><font color="Maroon" title="c11">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> = 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c8">ID</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">G2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">t</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_2.html#K2" title="BORSUK_2:func.2">-</a> <font color="Maroon" title="c8">ID</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">t</font>
<b>by </b><i><a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c8">ID</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#D5" target="_self" title="BORSUK_6:def.5">Def5</a></i>
<br/>.= 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">t</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:58_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="tmap_1.html#T91" title="TMAP_1:th.91">TMAP_1:91</a></i>
;<br/>
<b>hence </b><a NAME="E5:58_1"/>
<font color="Maroon" title="c11">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c12">s</font>,<font color="Maroon" title="c13">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c13">t</font>
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c12">F2</font> =  <a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as  <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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T39" title="YELLOW12:th.39">YELLOW12:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">f2</font> = <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">F2</font> as  <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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E9:58"/><i><font color="Green" title="E48">A5</font></i>: 
for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c13">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">s</font>, <font color="Maroon" title="c15">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:58_2"/>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E2:58_2"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">F2</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c13">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font> = 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font>
<b>by </b><i><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a></i>
;<br/><br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c14">f3</font> =  <a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as  <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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T39" title="YELLOW12:th.39">YELLOW12:39</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">f4</font> =  <a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as  <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 <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="yellow12.html#T40" title="YELLOW12:th.40">YELLOW12:40</a></i>;<br/>
<a NAME="E12:58"/><i><font color="Green" title="E49">A6</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c11">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T8" target="_self" title="BORSUK_6:th.8">Th8</a></i>;<br/>
<a NAME="E13:58"/><i><font color="Green" title="E50">A7</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c14">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T8" target="_self" title="BORSUK_6:th.8">Th8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">g1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E15:58"/><i><font color="Green" title="E51">A8</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c11">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Maroon" title="c13">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">r2</font> ) &amp; <font color="Maroon" title="c16">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E12:58"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="borsuk_6.html#T40" target="_self" title="BORSUK_6:th.40">Th40</a></i>;<br/>
<a NAME="E16:58"/><i><font color="Green" title="E52">A9</font></i>: 
for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">t</font>, <font color="Maroon" title="c18">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:58_3"/>
( <font color="Maroon" title="c11">f1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">t</font> &amp; <font color="Maroon" title="c13">f2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c17">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font> )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:58"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="borsuk_6.html#E9:58"><i><font color="Green" title="E48">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:58_3"/>
<font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c18">s</font>,<font color="Maroon" title="c17">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E15:58"><i><font color="Green" title="E51">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c17">g2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E18:58"/><i><font color="Green" title="E53">A10</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c14">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Maroon" title="c15">f4</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">r2</font> ) &amp; <font color="Maroon" title="c17">g2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E13:58"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="borsuk_6.html#T40" target="_self" title="BORSUK_6:th.40">Th40</a></i>;<br/>
<a NAME="E19:58"/><i><font color="Green" title="E54">A11</font></i>: 
for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">t</font>, <font color="Maroon" title="c19">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:58_4"/>
( <font color="Maroon" title="c14">f3</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font> &amp; <font color="Maroon" title="c15">f4</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> )
 
<b>by </b><i><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a>, <a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E2:58_4"/>
<font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c18">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c19">s</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E18:58"><i><font color="Green" title="E53">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:58"/>
for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>

<a NAME="E1:58_5"/>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
;<br/>
<a NAME="E2:58_5"/><b>then </b>
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">s</font>, <font color="Maroon" title="c20">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:58_5"/><i><font color="Green" title="E55">A12</font></i>: 
( <font color="Maroon" title="c19">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp; <font color="Maroon" title="c18">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c20">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">s</font> = <font color="Maroon" title="c19">s</font>, <font color="Maroon" title="c22">t</font> = <font color="Maroon" title="c20">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E4:58_5"><i><font color="Green" title="E55">A12</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">a</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_5_1"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">s</font> or <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c21">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_5_1_1"><b>suppose </b></a><a NAME="E1:58_5_1_1"/><i><font color="Green" title="E56">A13</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">s</font>
 ;<br/><div class="add"><a NAME="E2:58_5_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">t</font> &amp; <font color="Maroon" title="c22">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E3:58_5_1_1"/><b>then </b><i><font color="Green" title="E57">A14</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">s</font> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:58_5_1_1"><i><font color="Green" title="E56">A13</font></i></a>, <a class="ref" href="xreal_1.html#T174" title="XREAL_1:th.174">XREAL_1:174</a>, <a class="ref" href="xreal_1.html#T175" title="XREAL_1:th.175">XREAL_1:175</a></i>;<br/><i><font color="Green" title="E58">A15</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E16:58"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="borsuk_6.html#E4:58_5"><i><font color="Green" title="E55">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">s</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E19:58"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="borsuk_6.html#E4:58_5"><i><font color="Green" title="E55">A12</font></i></a></i>
;<br/><a NAME="E5:58_5_1_1"/><i><font color="Green" title="E59">A16</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E6:58_5_1_1"/>
<font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E7:58_5_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="borsuk_6.html#E3:58_5_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E8:58_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E3:58_5_1_1"><i><font color="Green" title="E57">A14</font></i></a>, <a class="txt" href="borsuk_6.html#E4:58_5_1_1"><i><font color="Green" title="E58">A15</font></i></a>, <a class="txt" href="borsuk_6.html#E5:58_5_1_1"><i><font color="Green" title="E59">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_5_1_2"><b>suppose </b></a><a NAME="E1:58_5_1_2"/><i><font color="Green" title="E56">A17</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c21">s</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c24">j</font> = 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font>;<br/><a NAME="E2:58_5_1_2"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <b>by </b><i><a class="ref" href="jordan5b.html#T4" title="JORDAN5B:th.4">JORDAN5B:4</a></i>;<br/><a NAME="E3:58_5_1_2"/><b>then </b>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font> &amp; 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/><a NAME="E4:58_5_1_2"/><b>then </b><i><font color="Green" title="E57">A18</font></i>: 
( <font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E1:58_5_1_2"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="xreal_1.html#T174" title="XREAL_1:th.174">XREAL_1:174</a>, <a class="ref" href="xreal_1.html#T175" title="XREAL_1:th.175">XREAL_1:175</a></i>;<br/><i><font color="Green" title="E58">A19</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E16:58"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="borsuk_6.html#E4:58_5"><i><font color="Green" title="E55">A12</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">s</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E19:58"><i><font color="Green" title="E54">A11</font></i></a>, <a class="txt" href="borsuk_6.html#E4:58_5"><i><font color="Green" title="E55">A12</font></i></a></i>
;<br/><a NAME="E6:58_5_1_2"/><i><font color="Green" title="E59">A20</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">s</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E7:58_5_1_2"/>
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E8:58_5_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c22">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:58_5_1_2"><i><font color="Green" title="E57">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E9:58_5_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">p</font></span>)</span> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E4:58_5_1_2"><i><font color="Green" title="E57">A18</font></i></a>, <a class="txt" href="borsuk_6.html#E5:58_5_1_2"><i><font color="Green" title="E58">A19</font></i></a>, <a class="txt" href="borsuk_6.html#E6:58_5_1_2"><i><font color="Green" title="E59">A20</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c18">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <b> such that </b><br/><a NAME="E22:58"/><i><font color="Green" title="E55">A21</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span><br/> for <font color="Olive" title="b2">r1</font>, <font color="Olive" title="b3">r2</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> &amp; <font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r2</font> holds <br/><font color="Maroon" title="c18">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b3">r2</font> ) &amp; <font color="Maroon" title="c18">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 <b>by </b><i><a class="txt" href="borsuk_6.html#E15:58"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="borsuk_6.html#E18:58"><i><font color="Green" title="E53">A10</font></i></a>, <a class="ref" href="borsuk_6.html#T41" target="_self" title="BORSUK_6:th.41">Th41</a></i>;<br/>
<a NAME="E23:58"/><i><font color="Green" title="E56">A22</font></i>: 
for <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c18">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Olive" title="b2">s</font>,<font color="Olive" title="b1">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">t</font>, <font color="Maroon" title="c20">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:58_6"/>
( <font color="Maroon" title="c16">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c20">s</font>,<font color="Maroon" title="c19">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font></span>)</span> &amp; <font color="Maroon" title="c17">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c20">s</font>,<font color="Maroon" title="c19">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">s</font> )
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E16:58"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="borsuk_6.html#E19:58"><i><font color="Green" title="E54">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:58_6"/>
<font color="Maroon" title="c18">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c20">s</font>,<font color="Maroon" title="c19">t</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c20">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E22:58"><i><font color="Green" title="E55">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:58"/><i><font color="Green" title="E57">A23</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">s</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="borsuk_6.html#E23:58"><i><font color="Green" title="E56">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E25:58"/><i><font color="Green" title="E58">A24</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>thus </b><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">s</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:58"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="borsuk_6.html#E23:58"><i><font color="Green" title="E56">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">s</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E26:58"/><i><font color="Green" title="E59">A25</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c20">oo</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c19">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">oo</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:58"><i><font color="Green" title="E56">A22</font></i></a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E27:58"/><i><font color="Green" title="E60">A26</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds  <font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Olive" title="b1">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon" title="c20">oo</font> = 1 as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c19">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c19">t</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">oo</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">t</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E23:58"><i><font color="Green" title="E56">A22</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E61">A27</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">h</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>
;<br/>
<b>set </b><font color="Maroon" title="c19">H</font> = <font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font>;<br/>
<a NAME="E29:58"/>
<font color="Maroon" title="c4">P</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/>
<a NAME="E30:58"/><b>then </b><i><font color="Green" title="E62">A28</font></i>: 
<font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E22:58"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E31:58"/><i><font color="Green" title="E63">A29</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:58_12"/>
( <font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E2:58_12"/><b>then </b><i><font color="Green" title="E64">A30</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">h</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:58"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:58_12"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E4:58_12"/><b>then </b><i><font color="Green" title="E65">A31</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">h</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:58"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:58_12"/>
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
;<br/>
<a NAME="E6:58_12"/><b>then </b><i><font color="Green" title="E66">A32</font></i>: 
<font color="Maroon" title="c20">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:58_12"><i><font color="Green" title="E64">A30</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E24:58"><i><font color="Green" title="E57">A23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E6:58_12"><i><font color="Green" title="E66">A32</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="borsuk_6.html#D6" target="_self" title="BORSUK_6:def.6">Def6</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">s</font>,1 = 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">s</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:58_12"><i><font color="Green" title="E65">A31</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">s</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E25:58"><i><font color="Green" title="E58">A24</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E32:58"/>
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/>

<a NAME="E1:58_13"/>
( <font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E2:58_13"/><b>then </b><i><font color="Green" title="E64">A33</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c20">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">h</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:58"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E3:58_13"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E4:58_13"/><b>then </b><i><font color="Green" title="E65">A34</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c20">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">h</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E28:58"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c20">t</font> = 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c20">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E2:58_13"><i><font color="Green" title="E64">A33</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="borsuk_6.html#E26:58"><i><font color="Green" title="E59">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c20">t</font> = 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<font color="Maroon" title="c20">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E4:58_13"><i><font color="Green" title="E65">A34</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="borsuk_6.html#E27:58"><i><font color="Green" title="E60">A26</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="borsuk_6.html#E1:58"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E33:58"/>
 <a href="borsuk_6.html#K2" title="BORSUK_6:func.2">RePar</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">P</font> <a href="borsuk_2.html#R3" title="BORSUK_2:pred.3">are_homotopic</a> 
 <b>by </b><i><a class="txt" href="borsuk_6.html#E30:58"><i><font color="Green" title="E62">A28</font></i></a>, <a class="txt" href="borsuk_6.html#E31:58"><i><font color="Green" title="E63">A29</font></i></a>, <a class="ref" href="borsuk_2.html#D7" title="BORSUK_2:def.7">BORSUK_2:def 7</a></i>;<br/>


</div>
