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<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>;<br/><b>let </b><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">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>, <font color="Maroon">Q</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>, <font color="Maroon">R</font> be    <a href="borsuk_2.html#M1">Path</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>







<b>assume </b><a NAME="E1:108"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="borsuk_2.html#R3">are_homotopic</a>  &amp; <font color="Maroon">Q</font>,<font color="Maroon">R</font> <a href="borsuk_2.html#R3">are_homotopic</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">f</font> being   <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><b> such that </b><br/><a NAME="E3:108"/><i><font color="Green">E33</font></i>: 
( <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">Q</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_2.html#D7">BORSUK_2:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <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><b> such that </b><br/><a NAME="E5:108"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">g</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">g</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> &amp; <font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">R</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">g</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">g</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_2.html#D7">BORSUK_2:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P1</font> =  <a href="treal_1.html#K4">P[01]</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T15">TREAL_1:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P2</font> =  <a href="treal_1.html#K4">P[01]</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>,<a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T15">TREAL_1:15</a></i>;<br/>
<b>set </b><font color="Maroon">e1</font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">e2</font> = <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<a NAME="E8:108"/><i><font color="Green">E35</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <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="E9:108"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:108"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> is  <a href="pre_topc.html#V5">continuous</a> <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> ,<span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<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_2.html#T12">BORSUK_2:12</a></i>;<br/>
<a NAME="E11:108"/><i><font color="Green">E62</font></i>: 
<span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> is  <a href="pre_topc.html#V5">continuous</a> <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> ,<span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>,<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_2.html#T12">BORSUK_2:12</a></i>;<br/>
<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span>;<br/>
<a NAME="E12:108"/><i><font color="Green">E63</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span> <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> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E13:108"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> <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="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4">P[01]</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1,<span class="p4">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span></span>)</span></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#T88">FUNCT_3:88</a></i>;<br/>
<a NAME="E14:108"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</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="txt" href="borsuk_6.html#E21:108"><i><font color="Green">E91</font></i></a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
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>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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 </b><i><font color="Green">E65</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span>
<b>by </b><i><a class="txt" href="borsuk_6.html#E25:108"><i><font color="Green">E92</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
<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 <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="txt" href="borsuk_6.html#E18:108"><i><font color="Green">E88</font></i></a>, <a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>
;<br/>
<a NAME="E17:108"/><i><font color="Green">E66</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> )
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E18:108"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> <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> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E19:108"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <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> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<a NAME="E20:108"/><b>then </b><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="mcart_1.html#K12">:]</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="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E21:108"/><i><font color="Green">E91</font></i>: 
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E22:108"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ewa</font> = <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span></span><a href="mcart_1.html#K12">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <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_6.html#T27" target="_self">Th27</a>, <a class="ref" href="borsuk_6.html#T28" target="_self">Th28</a>, <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E24:108"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> <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> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<a NAME="E25:108"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span><a href="mcart_1.html#K12">:]</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="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E26:108"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Ewa1</font> = <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span><a href="mcart_1.html#K12">:]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <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_6.html#T28" target="_self">Th28</a>, , <a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>set </b><font color="Maroon">T1</font> = <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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font>;<br/>
<b>set </b><font color="Maroon">T2</font> = <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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font>;<br/>
<b>reconsider </b><font color="Maroon">A01</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="txt" href="borsuk_6.html#E34:108"><i><font color="Green">E95</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B01</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<a NAME="E30:108"/>
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E31:108"/><b>then </b><i><font color="Green">E93</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B02</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a>, <a class="ref" href="treal_1.html#T1">TREAL_1:1</a></i>;<br/>
<a NAME="E33:108"/>
<font color="Maroon">A01</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#D3">PRE_TOPC:def 3</a></i>;<br/>
<a NAME="E34:108"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">A01</font>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T100">TSEP_1:100</a></i>;<br/>
<a NAME="E35:108"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B01</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/>
<a NAME="E36:108"/><b>then </b><i><font color="Green">E143</font></i>: 
<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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T29" target="_self">Th29</a>, <a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/>
<a NAME="E37:108"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1 <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B02</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/>
<a NAME="E38:108"/><b>then </b><i><font color="Green">E144</font></i>: 
<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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><a href="topmetr.html#K5">I[01]</a> ,<span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T29" target="_self">Th29</a>, <a class="ref" href="borsuk_3.html#T26">BORSUK_3:26</a></i>;<br/>
<a NAME="E39:108"/><i><font color="Green">E145</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ewa</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Ewa1</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then </b><i><font color="Green">E147</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font></span>)</span></span>)</span> = 
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#T120">ZFMISC_1:120</a></i>
<br/>.= 
<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#T83">BORSUK_1:83</a>, <a class="ref" href="treal_1.html#T2">TREAL_1:2</a></i>
<br/>.= 
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>
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <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="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font></span>)</span>,<font color="Maroon">T</font> <b>by </b><i>, , <a class="ref" href="borsuk_6.html#T30" target="_self">Th30</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><span class="p2"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font></span>)</span>,<font color="Maroon">T</font> <b>by </b><i>, <a class="txt" href="borsuk_6.html#E20:108"><i><font color="Green">E90</font></i></a>, <a class="ref" href="borsuk_6.html#T31" target="_self">Th31</a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E43:108"/><i><font color="Green">E148</font></i>: 
( 0 is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 is   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; 1 <a href="real_1.html#K6">/</a> 2 is   <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="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E44:108"/><b>then </b><i><font color="Green">E149</font></i>: 
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> is  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> is  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  )
 
<b>by </b><i><a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<a NAME="E45:108"/><b>then </b>
<font color="Maroon">Ewa</font> is  <a href="compts_1.html#V6">compact</a> <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_3.html#T27">BORSUK_3:27</a></i>;<br/>
<a NAME="E46:108"/><b>then </b><i><font color="Green">E150</font></i>: 
<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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<a NAME="E47:108"/>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span> is  <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T34" target="_self">Th34</a>, <a class="ref" href="borsuk_4.html#T49">BORSUK_4:49</a></i>;<br/>
<a NAME="E48:108"/><b>then </b>
<font color="Maroon">Ewa1</font> is  <a href="compts_1.html#V6">compact</a> <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_6.html#T35" target="_self">Th35</a>, <a class="ref" href="borsuk_3.html#T27">BORSUK_3:27</a></i>;<br/>
<a NAME="E49:108"/><b>then </b><i><font color="Green">E151</font></i>: 
<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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T12">COMPTS_1:12</a></i>;<br/>
<i><font color="Green">E153</font></i>: <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font></span>)</span></span>)</span> = 
<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2">(<span class="default"><span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K9">/\</a> <span class="p3"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T32" target="_self">Th32</a>, <a class="ref" href="zfmisc_1.html#T122">ZFMISC_1:122</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="topmetr2.html#T1">TOPMETR2:1</a></i>
;<br/>
<i><font color="Green">E154</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> = 
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> ,<span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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 <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></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/>
<i><font color="Green">E155</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> = 
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> ,<span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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 <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span></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/>
<a NAME="E53:108"/><i><font color="Green">E156</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> <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="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">P1</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E54:108"/><i><font color="Green">E157</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> <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="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">P2</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D9">FUNCT_3:def 9</a></i>;<br/>
<a NAME="E55:108"/><i><font color="Green">E158</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E56:108"/><b>then </b><i><font color="Green">E159</font></i>: 
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E57:108"/>
for <font color="Olive">p</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font></span>)</span></span>)</span> holds <br/><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:108_7"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><span class="p3"><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="pre_topc.html#K3">|</a> <font color="Maroon">Ewa1</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:108_7"/><i><font color="Green">E160</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T39" target="_self">Th39</a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E4:108_7"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 1 <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:108_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:108_7"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> 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="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E9:108_7"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_7_1"><b>proof </b></a><div class="add">

<a NAME="E1:108_7_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">x</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> 
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E2:108_7_1"/>
1 <a href="real_1.html#K6">/</a> 2 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y'</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> =  <a href="treal_1.html#K1">(#)</a> 0,1, <font color="Maroon">r2</font> = 0,1 <a href="treal_1.html#K2">(#)</a>  as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<b>set </b><font color="Maroon">t'</font> = 1 <a href="real_1.html#K6">/</a> 2;<br/>
<i><font color="Green">E234</font></i>: <font color="Maroon">P1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#T14">TREAL_1:14</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = 1 <a href="real_1.html#K6">/</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> <b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<i><font color="Green">E235</font></i>: <font color="Maroon">P2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K3">+</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#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#T14">TREAL_1:14</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>
;<br/>
<a NAME="E8:108_7_1"/><i><font color="Green">E236</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span>
 
<b>by </b><i>, , , <a class="ref" href="borsuk_6.html#T41" target="_self">Th41</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:108_7_1"/><i><font color="Green">E237</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T40" target="_self">Th40</a>, , , <a class="ref" href="borsuk_6.html#T41" target="_self">Th41</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T44" target="_self">Th44</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T44" target="_self">Th44</a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x'</font>,1
<b>by </b><i>, <a class="txt" href="borsuk_6.html#E46"><i><font color="Green">Lemma120</font></i></a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">Q</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x'</font>,0
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T43" target="_self">Th43</a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E11:108_7_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:108_7"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>
 ;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">h</font> being   <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><b> such that </b><br/><a NAME="E59:108"/><i><font color="Green">E238</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">g1</font> &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T33" target="_self">Th33</a>, <a class="ref" href="borsuk_6.html#T37" target="_self">Th37</a>, <a class="ref" href="borsuk_6.html#T38" target="_self">Th38</a>, <a class="ref" href="borsuk_2.html#T1">BORSUK_2:1</a></i>;<br/>
<a NAME="E60:108"/><i><font color="Green">E239</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E61:108"/><i><font color="Green">E240</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E62:108"/><i><font color="Green">E241</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">h</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">h</font> <a href="binop_1.html#K1">.</a> <font color="Olive">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Olive">s</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_8"><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/>

<a NAME="E1:108_8"/><i><font color="Green">E242</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E2:108_8"/><b>then </b>
not 0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E3:108_8"/><b>then </b><i><font color="Green">E243</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E31:108"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:108_8"/><i><font color="Green">E244</font></i>: 
0 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E5:108_8"/><b>then </b><i><font color="Green">E245</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,0</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r1</font> =  <a href="treal_1.html#K1">(#)</a> 0,1, <font color="Maroon">r2</font> = 0,1 <a href="treal_1.html#K2">(#)</a>  as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<b>set </b><font color="Maroon">t'</font> = 0;<br/>
<i><font color="Green">E246</font></i>: <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K3">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T44" target="_self">Th44</a>, <a class="ref" href="treal_1.html#T14">TREAL_1:14</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
;<br/>
<i><font color="Green">E247</font></i>: <font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0 = 
<font color="Maroon">f1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T43" target="_self">Th43</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0</span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,0
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a>, <a class="ref" href="borsuk_6.html#T45" target="_self">Th45</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
<b>by </b><i/>
;<br/>
<a NAME="E9:108_8"/><i><font color="Green">E248</font></i>: 
1 <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E46"><i><font color="Green">Lemma120</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E10:108_8"/><b>then </b><i><font color="Green">E249</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 
<b>by </b><i><a class="txt" href="borsuk_6.html#E31:108"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:108_8"/><i><font color="Green">E250</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">s</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_6.html#T40" target="_self">Th40</a>, <a class="ref" href="borsuk_6.html#T50" target="_self">Th50</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:108_8"/>
( 1 <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; 1 <a href="hidden.html#R1">=</a> 0,1 <a href="treal_1.html#K2">(#)</a>  &amp; 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1 <a href="treal_1.html#K2">(#)</a>  )
 
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>;<br/>
<a NAME="E13:108_8"/><b>then </b><i><font color="Green">E251</font></i>: 
<font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="treal_1.html#T16">TREAL_1:16</a></i>;<br/>
<font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1 = 
<font color="Maroon">g1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1</span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a>, <a class="ref" href="borsuk_6.html#D5" target="_self">Def5</a></i>
<br/>.= 
<font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E15:108_8"/>
( <font color="Maroon">h</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">h</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">s</font>,1 <a href="hidden.html#R1">=</a> <font color="Maroon">R</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="borsuk_6.html#T46" target="_self">Th46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E63:108"/><i><font color="Green">E252</font></i>: 
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><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E64:108"/>
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">h</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">h</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="108_9"><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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_9_1"/>
( <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2 or <font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_9_1_1"><b>suppose </b></a><a NAME="E1:108_9_1_1"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:108_9_1_1"/><i><font color="Green">E254</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E3:108_9_1_1"/><i><font color="Green">E255</font></i>: 
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> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E4:108_9_1_1"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 <b>by </b><i>, <a class="ref" href="borsuk_6.html#T43" target="_self">Th43</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E5:108_9_1_1"/><b>then </b><i><font color="Green">E256</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E31:108"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:108_9_1_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="borsuk_6.html#T43" target="_self">Th43</a>, <a class="ref" href="borsuk_6.html#T44" target="_self">Th44</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E7:108_9_1_1"/><b>then </b><i><font color="Green">E258</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">r1</font> =  <a href="treal_1.html#K1">(#)</a> 0,1, <font color="Maroon">r2</font> = 0,1 <a href="treal_1.html#K2">(#)</a>  as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/><a NAME="E9:108_9_1_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">t</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T46" target="_self">Th46</a>, , <a class="ref" href="treal_1.html#T14">TREAL_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>

;<br/><a NAME="E11:108_9_1_1"/><b>then </b><i><font color="Green">E260</font></i>: 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> is   <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="borsuk_6.html#T43" target="_self">Th43</a>, </i>;<br/><b>thus </b><font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font> = 
<font color="Maroon">f1</font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T45" target="_self">Th45</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T50" target="_self">Th50</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T50" target="_self">Th50</a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 0,<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/><a NAME="E13:108_9_1_1"/>
not <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T26" target="_self">Th26</a>, <a class="ref" href="borsuk_6.html#T43" target="_self">Th43</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E14:108_9_1_1"/><b>then </b><i><font color="Green">E261</font></i>: 
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E31:108"><i><font color="Green">E93</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E15:108_9_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,<span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T26" target="_self">Th26</a>, <a class="ref" href="borsuk_6.html#T43" target="_self">Th43</a>, <a class="ref" href="borsuk_6.html#T44" target="_self">Th44</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E16:108_9_1_1"/><b>then </b><i><font color="Green">E262</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span>
 <b>by </b><i>, <a class="txt" href="borsuk_6.html#E46"><i><font color="Green">Lemma120</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K5">-</a> 0</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T46" target="_self">Th46</a>, , <a class="ref" href="treal_1.html#T14">TREAL_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>

;<br/><a NAME="E18:108_9_1_1"/><b>then </b><i><font color="Green">E263</font></i>: 
<font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> is   <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="borsuk_6.html#T43" target="_self">Th43</a>, </i>;<br/><b>thus </b><font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font> = 
<font color="Maroon">f1</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P1</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> 1,<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_9_1_2"><b>suppose </b></a><a NAME="E1:108_9_1_2"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="real_1.html#K6">/</a> 2
 ;<br/><div class="add"><a NAME="E2:108_9_1_2"/><i><font color="Green">E265</font></i>: 
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> 
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E3:108_9_1_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/><a NAME="E4:108_9_1_2"/><b>then </b><i><font color="Green">E266</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span>,1</span>)</span>
 <b>by </b><i>, <a class="ref" href="borsuk_6.html#T43" target="_self">Th43</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E5:108_9_1_2"/><b>then </b><i><font color="Green">E267</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E31:108"><i><font color="Green">E93</font></i></a>, <a class="ref" href="borsuk_6.html#T44" target="_self">Th44</a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:108_9_1_2"/><i><font color="Green">E268</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T40" target="_self">Th40</a>, <a class="ref" href="borsuk_6.html#T44" target="_self">Th44</a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">r1</font> =  <a href="treal_1.html#K1">(#)</a> 0,1, <font color="Maroon">r2</font> = 0,1 <a href="treal_1.html#K2">(#)</a>  as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/><a NAME="E8:108_9_1_2"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">t</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T50" target="_self">Th50</a>, <a class="ref" href="treal_1.html#T14">TREAL_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/><a NAME="E10:108_9_1_2"/><b>then </b><i><font color="Green">E270</font></i>: 
<font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> is   <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="borsuk_6.html#T43" target="_self">Th43</a>, </i>;<br/><b>thus </b><font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font> = 
<font color="Maroon">g1</font> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T45" target="_self">Th45</a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> 0,<font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T46" target="_self">Th46</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 0</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T46" target="_self">Th46</a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> 0,<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, </i>
;<br/><a NAME="E12:108_9_1_2"/><i><font color="Green">E271</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g1</font>
 <b>by </b><i><a class="txt" href="borsuk_6.html#E31:108"><i><font color="Green">E93</font></i></a>, <a class="txt" href="borsuk_6.html#E46"><i><font color="Green">Lemma120</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E13:108_9_1_2"/><i><font color="Green">E272</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<font color="Maroon">t</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_6.html#T40" target="_self">Th40</a>, <a class="txt" href="borsuk_6.html#E46"><i><font color="Green">Lemma120</font></i></a>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#T50" target="_self">Th50</a>, <a class="ref" href="treal_1.html#T14">TREAL_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</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#K4">*</a> 0</span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1

;<br/><a NAME="E15:108_9_1_2"/><b>then </b><i><font color="Green">E273</font></i>: 
<font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> is   <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="borsuk_6.html#T43" target="_self">Th43</a>, </i>;<br/><b>thus </b><font color="Maroon">h</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font> = 
<font color="Maroon">g1</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="borsuk_2.html#K3">[:</a><span class="default"><span class="p3">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span>,<font color="Maroon">P2</font></span><a href="borsuk_2.html#K3">:]</a></span> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <a href="topmetr.html#K5">I[01]</a> </span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_3.html#T86">FUNCT_3:86</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> 1,<span class="p1">(<span class="default"><font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="borsuk_6.html#T34" target="_self">Th34</a>, <a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="borsuk_6.html#D6" target="_self">Def6</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E65:108"/>
<font color="Maroon">P</font>,<font color="Maroon">R</font> <a href="borsuk_2.html#R3">are_homotopic</a> 
 <b>by </b><i>, <a class="ref" href="borsuk_6.html#T41" target="_self">Th41</a>, <a class="ref" href="borsuk_2.html#D7">BORSUK_2:def 7</a></i>;<br/>


</div>
