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

<a NAME="E1:20_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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">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="c9">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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a>, <a class="ref" href="treal_1.html#T3" title="TREAL_1:th.3">TREAL_1:3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">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/>
<span class="kw">set </span><font color="Maroon" title="c11">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/>
<span class="kw">set </span><font color="Maroon" title="c12">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/>
<span class="kw">set </span><font color="Maroon" title="c13">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/>
<span class="kw">set </span><font color="Maroon" title="c14">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/>
<span class="lab"><font color="Green" title="E12">A3</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>
;<br/>
<span class="lab"><font color="Green" title="E13">A4</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>
;<br/>
<a NAME="E6:20_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 for <font color="Olive" title="b1">t9</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">t9</font> &amp; <font color="Olive" title="b1">t9</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">t9</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">t9</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="20_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c15">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="c16">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T5" title="TREAL_1:th.5">TREAL_1:5</a></span>;<br/>
<a NAME="E2:20_1_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E15">A6</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">t9</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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="c17">t9</font> &amp; <font color="Maroon" title="c17">t9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 implies <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="c17">t9</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="c17">t9</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:20_1_1_3"/>
( 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="c17">t9</font> &amp; <font color="Maroon" title="c17">t9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c17">t9</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="c17">t9</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span></span><br/>

<a NAME="E5:20_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c17">t9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1_1_3"><span class="lab"><font color="Green" title="E15">A6</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">s</font> = <font color="Maroon" title="c17">t9</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/>
<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="c18">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c15">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="c17">t9</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="c15">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="c16">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T11" title="TREAL_1:th.11">TREAL_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c17">t9</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T5" title="TREAL_1:th.5">TREAL_1:5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:20_1_1_3"/>
<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="c17">t9</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="c17">t9</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1_3"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c15">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="c9">T2</font>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">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="c8">T1</font>,<font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></span>;<br/>
<a NAME="E9:20_1_1"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
(  <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> &amp;  <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T12" title="TREAL_1:th.12">TREAL_1:12</a></span>;<br/>
<a NAME="E10:20_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#K2" title="RELSET_1:func.2">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#K2" title="RELSET_1:func.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></span>;<br/>
<a NAME="E11:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A9</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:20_1_1"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
(  <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  is  <a href="pre_topc.html#V8" title="PRE_TOPC:attr.8">T_2</a>  &amp; <font color="Maroon" title="c8">T1</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="heine.html#T4" title="HEINE:th.4">HEINE:4</a>, <a class="ref" href="pcomps_1.html#T34" title="PCOMPS_1:th.34">PCOMPS_1:34</a>, <a class="ref" href="topmetr.html#D6" title="TOPMETR:def.6">TOPMETR:def 6</a></span>;<br/>
<a NAME="E13:20_1_1"/><span class="lab"><font color="Green" title="E18">A11</font></span>: 
 for <font color="Olive" title="b1">t9</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">t9</font> &amp; <font color="Olive" title="b1">t9</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">t9</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">t9</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c17">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="c18">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T5" title="TREAL_1:th.5">TREAL_1:5</a></span>;<br/>
<a NAME="E2:20_1_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E19">A12</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>
<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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c19">t9</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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="c19">t9</font> &amp; <font color="Maroon" title="c19">t9</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 <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="c19">t9</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="c19">t9</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:20_1_1_4"/>
(  <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="c19">t9</font> &amp; <font color="Maroon" title="c19">t9</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 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c19">t9</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="c19">t9</font></span>)</span></span><br/>

<a NAME="E5:20_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A13</font></span>: 
<font color="Maroon" title="c19">t9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="kw">by</span> <span class="lab"><a class="txt" href="#E3:20_1_1_4"><span class="lab"><font color="Green" title="E19">A12</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">s</font> = <font color="Maroon" title="c19">t9</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/>
<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="c20">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">r2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c17">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="c19">t9</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="c17">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="c18">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T11" title="TREAL_1:th.11">TREAL_1:11</a></span>
<br/>.= 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c19">t9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="treal_1.html#T5" title="TREAL_1:th.5">TREAL_1:5</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:20_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="c19">t9</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="c19">t9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1_4"><span class="lab"><font color="Green" title="E20">A13</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="lab"><font color="Green" title="E19">A14</font></span>: <font color="Maroon" title="c16">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>

<br/>.= 
<font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></span>
<br/>.= 
<font color="Maroon" title="c15">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">pol</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a></span>
;<br/>
<a NAME="E15:20_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A15</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">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#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></span>;<br/>
<a NAME="E17:20_1_1"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">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> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E18:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a>, <a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></span>;<br/>
<a NAME="E19:20_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="20_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1_1_6"/>
 <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_1_1_6"/><span class="kw">then </span>
 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/>
<span class="kw">hence </span><a NAME="E3:20_1_1_6"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:20_1_1"/><span class="kw">then </span>
 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#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E18:20_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E22">A17</font></span>: <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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1_1"><span class="lab"><font color="Green" title="E18">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E23">A18</font></span>:  <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#K1" title="RELSET_1:func.1">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#K1" title="RELSET_1:func.1">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E16:20_1_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E18:20_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T174" title="XXREAL_1:th.174">XXREAL_1:174</a></span>
;<br/>
<a NAME="E23:20_1_1"/>
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c8">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> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c9">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>;<br/>
<a NAME="E24:20_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A19</font></span>: 
( <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c8">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="c9">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> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c8">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="c9">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="c7">pol</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_1.html#T40" title="BORSUK_1:th.40">BORSUK_1:40</a>, <a class="ref" href="xxreal_1.html#T174" title="XXREAL_1:th.174">XXREAL_1:174</a>, <a class="ref" href="xxreal_1.html#T418" title="XXREAL_1:th.418">XXREAL_1:418</a></span>;<br/>
<a NAME="E25:20_1_1"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
<font color="Maroon" title="c9">T2</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="heine.html#T4" title="HEINE:th.4">HEINE:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:20_1_1"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="txt" href="#E11:20_1_1"><span class="lab"><font color="Green" title="E16">A9</font></span></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#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E27:20_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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">f</font> = <font color="Maroon" title="c17">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:20_1_1"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E14:20_1_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E24:20_1_1"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E12:20_1_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="txt" href="#E25:20_1_1"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="ref" href="compts_1.html#T20" title="COMPTS_1:th.20">COMPTS_1:20</a>, <a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></span>;<br/>
<a NAME="E29:20_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="E30:20_1_1"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E18:20_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E26">A21</font></span>: <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
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></span>
;<br/>
<a NAME="E32:20_1_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="borsuk_2.html#D1" target="_self" title="BORSUK_2:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:20_1_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E31:20_1_1"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="ref" href="borsuk_2.html#D2" target="_self" title="BORSUK_2:def.2">Def2</a></span>;<br/>
<a NAME="E34:20_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="20_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <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> ) &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="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> ) )</span><br/>

<a NAME="E1:20_1_1_10"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
<font color="Maroon" title="c20">t</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<a NAME="E2:20_1_1_10"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="borsuk_2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:20_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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1_1_10_1"/><span class="lab"><font color="Green" title="E29">A24</font></span>: 
<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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:20_1_1_10_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1_10"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E1:20_1_1_10"><span class="lab"><font color="Green" title="E27">A22</font></span></a></span>;<br/>
<a NAME="E3:20_1_1_10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A25</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_10_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_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="20_1_1_10_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_10_1_1_1"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
<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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:20_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#K1" title="RELSET_1:func.1">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="20_1_1_10_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_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#K1" title="RELSET_1:func.1">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:20_1_1_10_1_1_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E18:20_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E3:20_1_1_10_1"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:20_1_1_10_1_1_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T418" title="XXREAL_1:th.418">XXREAL_1:418</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:20_1_1_10_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_10_1_1_1"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1_1"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E2:20_1_1_10"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E1:20_1_1_10"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E1:20_1_1_10_1"><span class="lab"><font color="Green" title="E29">A24</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E4:20_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_10_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_10_1_1_2"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:20_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="E3:20_1_1_10_1_1_2"/><span class="kw">then </span>
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/><a NAME="E4:20_1_1_10_1_1_2"/><span class="kw">then </span>
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" title="TOPMETR:th.18">TOPMETR:18</a></span>;<br/><a NAME="E5:20_1_1_10_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_10_1_1_2"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="topmetr.html#T20" title="TOPMETR:th.20">TOPMETR:20</a></span>;<br/><span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:20_1_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E1:20_1_1_10_1_1_2"><span class="lab"><font color="Green" title="E31">A27</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:20_1_1"><span class="lab"><font color="Green" title="E18">A11</font></span></a>, <a class="txt" href="#E2:20_1_1_10"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E1:20_1_1_10"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E1:20_1_1_10_1"><span class="lab"><font color="Green" title="E29">A24</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E7:20_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<a NAME="E4:20_1_1_10"/><span class="lab"><font color="Green" title="E29">A28</font></span>: 
<font color="Maroon" title="c20">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="borsuk_2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:20_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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_10_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1_1_10_2"/><span class="lab"><font color="Green" title="E30">A29</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<a NAME="E2:20_1_1_10_2"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20_1_1_10"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="txt" href="#E1:20_1_1_10"><span class="lab"><font color="Green" title="E27">A22</font></span></a></span>;<br/>
<a NAME="E3:20_1_1_10_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#D1" title="RCOMP_1:def.1">RCOMP_1:def 1</a></span>;<br/>
<span class="kw">then </span><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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E18:20_1_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E4:20_1_1_10"><span class="lab"><font color="Green" title="E29">A28</font></span></a>, <a class="txt" href="#E1:20_1_1_10"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E1:20_1_1_10_2"><span class="lab"><font color="Green" title="E30">A29</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:20_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E35:20_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> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
