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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>, <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">P</font> be    <a href="borsuk_2.html#M1" target="_self">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="borsuk_2.html#R2" target="_self">are_connected</a> 
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="mcart_1.html#K1">`1</a> </span>)</span>;<br/>
<a NAME="E2:44"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
<font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E3:44_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:44"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i>
<a NAME="E5:44"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>,<font color="Maroon">T</font> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D7" target="_self">BORSUK_2:def 7</a><br/>

<a NAME="E8:44"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">W</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span><br/> for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">W</font>  ex <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">W</font> st <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font>;<br/>



<a NAME="E1:44_2"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E2:44_2"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W</font> <a href="mcart_1.html#K1">`1</a>  as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G'</font> = <font color="Maroon">G</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W1</font> <b>by </b><i>, </i>;<br/>
<a NAME="E5:44_2"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W1</font><b> such that </b><br/><a NAME="E7:44_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G'</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>
<a NAME="E8:44_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">W1</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">HH</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<a NAME="E9:44_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T118">ZFMISC_1:118</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">HH</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">H</font>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<a NAME="E11:44_2"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">AI</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  ;<br/>
<a NAME="E13:44_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">AI</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
;<br/>
<a NAME="E14:44_2"/><i><font color="Green">E53</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">HH</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">AI</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T47">BORSUK_1:47</a></i>;<br/>
<a NAME="E15:44_2"/><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E16:44_2"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">AI</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1"><i><font color="Green">Lemma30</font></i></a>, <a class="ref" href="tops_1.html#T43">TOPS_1:43</a></i>;<br/>
<a NAME="E17:44_2"/><b>then </b>
<font color="Maroon">W</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">AI</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E18:44_2"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">HH</font>
 
<b>by </b><i>, , <a class="txt" href="borsuk_2.html#E1:44"><i><font color="Green">E32</font></i></a>, <a class="ref" href="mcart_1.html#T11">MCART_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">HH</font> = <font color="Maroon">HH</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">HH</font>
;<br/>

<a NAME="E20:44_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">HH</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">HH</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_2_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">HH</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:44_2_1"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a> </span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i><a class="txt" href="borsuk_2.html#E2:44"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E6:44_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:44_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">b</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">b</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E2:44"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E8:44_2_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="txt" href="borsuk_2.html#E2:44"><i><font color="Green">E33</font></i></a>, </i>;<br/>
<a NAME="E9:44_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:44_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="borsuk_2.html#E2:44"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:44_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">HH</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E9:44"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">s</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>reconsider </b><font color="Maroon">s0</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,0</span><a href="domain_1.html#K1">]</a></span>, <font color="Maroon">s1</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">s</font>,1</span><a href="domain_1.html#K1">]</a></span> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E2:44_3"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:44_3"/><b>then </b>
( <font color="Maroon">s0</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">s1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:44_3"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s0</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s0</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s1</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s1</font>] )
 
<b>by </b><i/>;<br/>
<a NAME="E5:44_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E6:44_3"/>
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E7:44_3"/>
( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E10:44"/>
for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/>

<b>set </b><font color="Maroon">t0</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<b>set </b><font color="Maroon">t1</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E1:44_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:44_4"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:44_4"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:44_4"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:44_4"/><b>then </b>
( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="domain_1.html#K1">]</a></span> <a href="mcart_1.html#K1">`1</a> </span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E6:44_4"/>
( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:44"/>
( <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a> &amp; ( for <font color="Olive">s</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">t</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  holds <br/> ( <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 0,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 1,<font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) ) ) ) )
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>


</div>
