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

<b>set </b><font color="Maroon" title="c7">e1</font> =  <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">e2</font> =  <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">E1</font> = <font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">E2</font> = <font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span>;<br/>
<i><font color="Green" title="E12">A3</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</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="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>
;<br/>
<i><font color="Green" title="E13">A4</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span>)</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="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>
;<br/>
<a NAME="E3:21_1_1"/><i><font color="Green" title="E14">A5</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:21_1_1"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">P</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E5:21_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 
;<br/>
<a NAME="E6:21_1_1"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E3:21_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<i><font color="Green" title="E17">A8</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21_1_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="borsuk_2.html#E2:21_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="borsuk_2.html#E4:21_1_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="borsuk_2.html#E6:21_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
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#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T50" title="RCOMP_1:th.50">RCOMP_1:50</a></i>
;<br/>
<a NAME="E8:21_1_1"/><i><font color="Green" title="E18">A9</font></i>: 
for <font color="Olive" title="b1">t'</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <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="Olive" title="b1">t'</font> &amp; <font color="Olive" title="b1">t'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">t'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">t'</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_1_1_4"/><i><font color="Green" title="E19">A10</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="c12">t'</font> &amp; <font color="Maroon" title="c12">t'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 ;<br/>

<a NAME="E2:21_1_1_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</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/>
<b>then </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>
;<br/>
<a NAME="E4:21_1_1_4"/><b>then </b><i><font color="Green" title="E20">A11</font></i>: 
<font color="Maroon" title="c12">t'</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> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21_1_1_4"><i><font color="Green" title="E19">A10</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c12">t'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r1</font> =  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <font color="Maroon" title="c15">r2</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a></i>;<br/>
<span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c14">r1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <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></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">t'</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c14">r1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<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="c15">r2</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#T14" title="TREAL_1:th.14">TREAL_1:14</a></i>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">t'</font>
<b>by </b><i><a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a></i>
;<br/>
<b>hence </b><a NAME="E8:21_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c12">t'</font></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E4:21_1_1_4"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:21_1_1"/>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_5"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/>

<a NAME="E2:21_1_1_5"/><b>then </b>
 ex <font color="Olive" title="b1">rr</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">rr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">rr</font> &amp; <font color="Olive" title="b1">rr</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
;<br/>
<b>hence </b><a NAME="E3:21_1_1_5"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:21_1_1"/><b>then </b>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:21_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="borsuk_2.html#E6:21_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E19">A12</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <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="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E8:21_1_1"><i><font color="Green" title="E18">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></i>
;<br/>
<a NAME="E12:21_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/>
<a NAME="E13:21_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p4">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> as   <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> ,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="borsuk_2.html#E7:21_1_1"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">T1</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>, <font color="Maroon" title="c14">T2</font> =  <a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1 as    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T6" title="TREAL_1:th.6">TREAL_1:6</a></i>;<br/>
<a NAME="E16:21_1_1"/><i><font color="Green" title="E20">A13</font></i>: 
 <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> is  <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">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>
 
<b>by </b><i><a class="ref" href="treal_1.html#T15" title="TREAL_1:th.15">TREAL_1:15</a></i>;<br/>
<a NAME="E17:21_1_1"/><i><font color="Green" title="E21">A14</font></i>: 
 <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="treal_1.html#T15" title="TREAL_1:th.15">TREAL_1:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">ff</font> = <font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c13">T1</font>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E19:21_1_1"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">gg</font> = <font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c14">T2</font>,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">FUNCT_2:19</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E21:21_1_1"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c17">pol</font> = 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 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#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E23:21_1_1"/>
( <font color="Maroon" title="c5">P</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c6">Q</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="borsuk_2.html#E2:21"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></i>;<br/>
<a NAME="E24:21_1_1"/><b>then </b><i><font color="Green" title="E23">A16</font></i>: 
( <font color="Maroon" title="c15">ff</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c16">gg</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E16:21_1_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="txt" href="borsuk_2.html#E17:21_1_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E25:21_1_1"/><i><font color="Green" title="E24">A17</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c13">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E26:21_1_1"/><i><font color="Green" title="E25">A18</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c14">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E27:21_1_1"/><b>then </b><i><font color="Green" title="E26">A19</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c13">T1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c14">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E25:21_1_1"><i><font color="Green" title="E24">A17</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#T50" title="RCOMP_1:th.50">RCOMP_1:50</a></i>;<br/>
<a NAME="E28:21_1_1"/><i><font color="Green" title="E27">A20</font></i>: 
for <font color="Olive" title="b1">t'</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t'</font> &amp; <font color="Olive" title="b1">t'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">t'</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">t'</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:21_1_1_7"/><i><font color="Green" title="E28">A21</font></i>: 
( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">t'</font> &amp; <font color="Maroon" title="c18">t'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:21_1_1_7"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span>)</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/>
<b>then </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> = 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>
;<br/>
<a NAME="E4:21_1_1_7"/><b>then </b><i><font color="Green" title="E29">A22</font></i>: 
<font color="Maroon" title="c18">t'</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> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21_1_1_7"><i><font color="Green" title="E28">A21</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">s</font> = <font color="Maroon" title="c18">t'</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">r1</font> =  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1, <font color="Maroon" title="c21">r2</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a></i>;<br/>
<span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c20">r1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">t'</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">r1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c21">r2</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <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#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#T14" title="TREAL_1:th.14">TREAL_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">t'</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<b>by </b><i><a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a></i>
;<br/>
<b>hence </b><a NAME="E8:21_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c18">t'</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E4:21_1_1_7"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E28">A23</font></i>: <font color="Maroon" title="c15">ff</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#K10" title="REAL_1:func.10">/</a> 2</span>)</span> = 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E8:21_1_1"><i><font color="Green" title="E18">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:21"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c16">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">pol</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E28:21_1_1"><i><font color="Green" title="E27">A20</font></i></a></i>
;<br/>
<a NAME="E30:21_1_1"/><i><font color="Green" title="E29">A24</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c13">T1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c14">T2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">pol</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E25:21_1_1"><i><font color="Green" title="E24">A17</font></i></a>, <a class="txt" href="borsuk_2.html#E26:21_1_1"><i><font color="Green" title="E25">A18</font></i></a>, <a class="ref" href="topmetr2.html#T1" title="TOPMETR2:th.1">TOPMETR2:1</a></i>;<br/>
<a NAME="E31:21_1_1"/>
 <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38" title="PCOMPS_1:th.38">PCOMPS_1:38</a>, <a class="ref" href="topmetr.html#D7" title="TOPMETR:def.7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E32:21_1_1"/><b>then </b>
( <font color="Maroon" title="c13">T1</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp; <font color="Maroon" title="c14">T2</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> &amp;  <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a>  &amp; <font color="Maroon" title="c15">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">pol</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">pol</font> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E29:21_1_1"><i><font color="Green" title="E28">A23</font></i></a>, <a class="ref" href="heine.html#T11" title="HEINE:th.11">HEINE:11</a>, <a class="ref" href="topmetr.html#T3" title="TOPMETR:th.3">TOPMETR:3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c12">f</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="borsuk_2.html#E24:21_1_1"><i><font color="Green" title="E23">A16</font></i></a>, <a class="txt" href="borsuk_2.html#E27:21_1_1"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="borsuk_2.html#E30:21_1_1"><i><font color="Green" title="E29">A24</font></i></a>, <a class="ref" href="topmetr2.html#T4" title="TOPMETR2:th.4">TOPMETR2:4</a></i>;<br/>
<a NAME="E34:21_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E35:21_1_1"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:21_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="borsuk_2.html#E6:21_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E30">A25</font></i>: <font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E28:21_1_1"><i><font color="Green" title="E27">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:21"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></i>
;<br/>
<a NAME="E37:21_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E11:21_1_1"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="borsuk_2.html#D1" target="_self" title="BORSUK_2:def.1">Def1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c19">f</font> = <font color="Maroon" title="c18">f</font> as    <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="c4">c</font> <b>by </b><i><a class="txt" href="borsuk_2.html#E11:21_1_1"><i><font color="Green" title="E19">A12</font></i></a>, <a class="txt" href="borsuk_2.html#E36:21_1_1"><i><font color="Green" title="E30">A25</font></i></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></i>;<br/>
<a NAME="E39:21_1_1"/>
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/> ( ( <font color="Olive" title="b1">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 implies <font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">t</font></span>)</span> ) &amp; ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> implies <font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_10"><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="E1:21_1_1_10"/><i><font color="Green" title="E31">A26</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="c20">t</font> &amp; <font color="Maroon" title="c20">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E2:21_1_1_10"/><i><font color="Green" title="E32">A27</font></i>: 
<font color="Maroon" title="c20">t</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:21_1_1_10"/>
( <font color="Maroon" title="c20">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 implies <font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_10_1"/><i><font color="Green" title="E33">A28</font></i>: 
<font color="Maroon" title="c20">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ;<br/>

<a NAME="E2:21_1_1_10_1"/><b>then </b>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : (  <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="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21_1_1_10"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="borsuk_2.html#E2:21_1_1_10"><i><font color="Green" title="E32">A27</font></i></a></i>;<br/>
<a NAME="E3:21_1_1_10_1"/><b>then </b><i><font color="Green" title="E34">A29</font></i>: 
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_10_1_1"/>
( <font color="Maroon" title="c20">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:21_1_1_10_1_1_1"/><i><font color="Green" title="E35">A30</font></i>: 
<font color="Maroon" title="c20">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ;<br/><div class="add"><a NAME="E2:21_1_1_10_1_1_1"/>
not <font color="Maroon" title="c20">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_10_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_10_1_1_1_1"/>
<font color="Maroon" title="c20">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:21_1_1_10_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:21_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="borsuk_2.html#E6:21_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="borsuk_2.html#E3:21_1_1_10_1"><i><font color="Green" title="E34">A29</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:21_1_1_10_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="topmetr2.html#T1" title="TOPMETR2:th.1">TOPMETR2:1</a></i>;<br/>
<b>hence </b><a NAME="E4:21_1_1_10_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="borsuk_2.html#E1:21_1_1_10_1_1_1"><i><font color="Green" title="E35">A30</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><b>then </b><font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">t</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E8:21_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="borsuk_2.html#E2:21_1_1_10"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10_1"><i><font color="Green" title="E33">A28</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:21_1_1_10_1_1_1"/>
<font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:21_1_1_10_1_1_2"/><i><font color="Green" title="E35">A31</font></i>: 
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
 ;<br/><div class="add"><a NAME="E2:21_1_1_10_1_1_2"/>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 ;<br/><a NAME="E3:21_1_1_10_1_1_2"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:21_1_1_10_1_1_2"/><b>then </b>
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <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">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/><a NAME="E5:21_1_1_10_1_1_2"/><b>then </b>
<font color="Maroon" title="c20">t</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="borsuk_2.html#E19:21_1_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10_1_1_2"><i><font color="Green" title="E35">A31</font></i></a>, <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="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">t</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E29:21_1_1"><i><font color="Green" title="E28">A23</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10_1_1_2"><i><font color="Green" title="E35">A31</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E8:21_1_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="borsuk_2.html#E2:21_1_1_10"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10_1"><i><font color="Green" title="E33">A28</font></i></a></i>
;<br/><b>hence </b><a NAME="E7:21_1_1_10_1_1_2"/>
<font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>thus </b><a NAME="E4:21_1_1_10"/>
( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">t</font> implies <font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_10_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_10_2"/><i><font color="Green" title="E33">A32</font></i>: 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">t</font>
 ;<br/>

<a NAME="E2:21_1_1_10_2"/><b>then </b>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> : ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:21_1_1_10"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="borsuk_2.html#E2:21_1_1_10"><i><font color="Green" title="E32">A27</font></i></a></i>;<br/>
<a NAME="E3:21_1_1_10_2"/><b>then </b>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></i>;<br/>
<b>then </b><font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span>,1,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span>,<span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">t</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:21_1_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="borsuk_2.html#E6:21_1_1"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E28:21_1_1"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10"><i><font color="Green" title="E31">A26</font></i></a>, <a class="txt" href="borsuk_2.html#E2:21_1_1_10"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="borsuk_2.html#E1:21_1_1_10_2"><i><font color="Green" title="E33">A32</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:21_1_1_10_2"/>
<font color="Maroon" title="c19">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c20">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E40:21_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <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="c4">c</font> st <br/>for <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 <br/> ( ( <font color="Olive" title="b2">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">t</font></span>)</span> ) &amp; ( 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">t</font> implies <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</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="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> ) )
 ;<br/>


</div>
