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

<b>let </b><font color="Maroon" title="c6">P</font>, <font color="Maroon" title="c7">Q</font> be    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>

<b>given </b><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E1:43_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c8">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; ( for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; ( 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="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) ) ) ) )
 ;<br/>

<a NAME="E2:43_1_1"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="grcat_1.html#D11" title="GRCAT_1:def.11">GRCAT_1:def 11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">fA</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">fB</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span> 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> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T11" title="TREAL_1:th.11">TREAL_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">F</font> = <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>;<br/>
<a NAME="E5:43_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
<span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#T12" target="_self" title="BORSUK_2:th.12">Th12</a></i>;<br/>
<a NAME="E6:43_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">LL</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">ff</font> = <font color="Maroon" title="c8">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E8:43_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c13">ff</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="borsuk_2.html#E5:43_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E9:43_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:43_1_1_1"/><i><font color="Green" title="E27">A6</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <br/> for <font color="Olive" title="b2">t'</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t'</font> holds <br/><span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b2">t'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/><b>let </b><font color="Maroon" title="c16">t'</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_1_1_1_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">t'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c17">ee</font> = <font color="Maroon" title="c15">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E3:43_1_1_1_1"/><i><font color="Green" title="E29">A8</font></i>: 
( 0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="treal_1.html#D1" title="TREAL_1:def.1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2" title="TREAL_1:def.2">TREAL_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t</font> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">ee</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c16">t'</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_1_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="txt" href="borsuk_2.html#E3:43_1_1_1_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="treal_1.html#T10" title="TREAL_1:th.10">TREAL_1:10</a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c16">t'</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:43_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c16">t'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:43_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:43_1_1_1"/><b>then </b><i><font color="Green" title="E28">A9</font></i>: 
( <font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span> &amp; 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E6:43_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="borsuk_2.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E29">A10</font></i>: <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,0 = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">s</font>,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

;<br/>
<i><font color="Green" title="E30">A11</font></i>: <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,1 = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E3:43_1_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="borsuk_1.html#K22" title="BORSUK_1:func.22">1[01]</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">s</font>,<span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_1"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">s</font>,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>

;<br/>
<a NAME="E6:43_1_1_1"/><i><font color="Green" title="E31">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E7:43_1_1_1"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">s</font>,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E3:43_1_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E8:43_1_1_1"/><i><font color="Green" title="E33">A14</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">s</font>,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E3:43_1_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="borsuk_2.html#E6:43_1_1_1"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green" title="E34">A15</font></i>: <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,0 = 
<font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,1
<b>by </b><i><a class="txt" href="borsuk_2.html#E8:43_1_1_1"><i><font color="Green" title="E33">A14</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="borsuk_2.html#E4:43_1_1_1"><i><font color="Green" title="E29">A10</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>
;<br/>
<font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,1 = 
<font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,0
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_1"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="borsuk_2.html#E5:43_1_1_1"><i><font color="Green" title="E30">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E11:43_1_1_1"/>
( <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font> &amp; <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s</font> )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E9:43_1_1_1"><i><font color="Green" title="E34">A15</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:43_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="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:43_1_1_2"/><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
;<br/>
<a NAME="E2:43_1_1_2"/><i><font color="Green" title="E28">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">tt</font> = <font color="Maroon" title="c14">t</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_2"><i><font color="Green" title="E27">A16</font></i></a>, <a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a></i>;<br/>
<a NAME="E4:43_1_1_2"/>
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> <br/> for <font color="Olive" title="b2">t'</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t'</font> holds <br/><span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Olive" title="b2">t'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">t</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ;<br/><b>let </b><font color="Maroon" title="c17">t'</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E1:43_1_1_2_1"/><i><font color="Green" title="E29">A18</font></i>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c18">ee</font> = <font color="Maroon" title="c16">t</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E3:43_1_1_2_1"/><i><font color="Green" title="E30">A19</font></i>: 
( 0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="treal_1.html#D1" title="TREAL_1:def.1">TREAL_1:def 1</a>, <a class="ref" href="treal_1.html#D2" title="TREAL_1:def.2">TREAL_1:def 2</a></i>;<br/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">t</font> = 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">ee</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c17">t'</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1_2_1"><i><font color="Green" title="E29">A18</font></i></a>, <a class="txt" href="borsuk_2.html#E3:43_1_1_2_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="treal_1.html#T10" title="TREAL_1:th.10">TREAL_1:10</a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c17">t'</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E5:43_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c17">t'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:43_1_1_2"/><b>then </b><i><font color="Green" title="E29">A20</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c15">tt</font>
 
;<br/>
<a NAME="E6:43_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</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 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:43_1_1_2"/><b>then </b><i><font color="Green" title="E30">A21</font></i>: 
( 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span> &amp; <font color="Maroon" title="c14">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:43_1_1_2"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="borsuk_2.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E31">A22</font></i>: <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c14">t</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c15">tt</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E5:43_1_1_2"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<i><font color="Green" title="E32">A23</font></i>: <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c14">t</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c15">tt</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="borsuk_2.html#E5:43_1_1_2"><i><font color="Green" title="E29">A20</font></i></a>, <a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c16">t'</font> = 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c15">tt</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_2.html#E18"><i><font color="Green" title="E14">Lm5</font></i></a></i>;<br/>
<a NAME="E11:43_1_1_2"/><i><font color="Green" title="E33">A24</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9" title="FUNCT_3:def.9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E12:43_1_1_2"/><b>then </b><i><font color="Green" title="E34">A25</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<font color="Maroon" title="c14">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:43_1_1_2"/><i><font color="Green" title="E35">A26</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">0,<font color="Maroon" title="c14">t</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">[:</a><span class="default"><font color="Maroon" title="c9">fA</font>,<font color="Maroon" title="c10">fB</font></span><a href="borsuk_2.html#K3" title="BORSUK_2:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E7:43_1_1_2"><i><font color="Green" title="E30">A21</font></i></a>, <a class="txt" href="borsuk_2.html#E11:43_1_1_2"><i><font color="Green" title="E33">A24</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<i><font color="Green" title="E36">A27</font></i>: <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c14">t</font> = 
<font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c16">t'</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E13:43_1_1_2"><i><font color="Green" title="E35">A26</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="borsuk_2.html#E8:43_1_1_2"><i><font color="Green" title="E31">A22</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>
;<br/>
<font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c14">t</font> = 
<font color="Maroon" title="c8">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c16">t'</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E12:43_1_1_2"><i><font color="Green" title="E34">A25</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="txt" href="borsuk_2.html#E9:43_1_1_2"><i><font color="Green" title="E32">A23</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:43_1_1"><i><font color="Green" title="E22">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E16:43_1_1_2"/>
( <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c13">ff</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E14:43_1_1_2"><i><font color="Green" title="E36">A27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:43_1_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> </span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>,<font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; ( for <font color="Olive" title="b2">s</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  holds <br/> ( <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">s</font>,0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">s</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">s</font> &amp; ( for <font color="Olive" title="b3">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">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 0,<font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> 1,<font color="Olive" title="b3">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) ) ) ) )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E8:43_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="borsuk_2.html#E9:43_1_1"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>


</div>
