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<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> , <font color="Maroon">b</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D3" target="_self">BORSUK_2:def 3</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_1"/>
( <font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> or <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></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/>;<br/><a NAME="E3:49_1_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b</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="E4:49_1_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/><a NAME="E5:49_1_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2">(#)</a> </span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T11">TREAL_1:11</a></i>;<br/><a NAME="E6:49_1_1_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></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="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2">(#)</a> </span>)</span> as   <a href="funct_2.html#NM1">Function</a> of 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>  <b>by </b><i>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a>  ;<br/><b>take </b>
<font color="Maroon">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1" target="_self">BORSUK_2:def 1</a><br/><b>thus </b><a NAME="E9:49_1_1_1_1"/>
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><a NAME="E10:49_1_1_1_1"/>
0 <a href="hidden.html#R1">=</a>  <a href="treal_1.html#K1">(#)</a> 0,1
 <b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>;<br/><b>hence </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 = 
 <a href="treal_1.html#K1">(#)</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
;<br/><br/><a NAME="E12:49_1_1_1_1"/>
1 <a href="hidden.html#R1">=</a> 0,1 <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/><b>hence </b><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="treal_1.html#K2">(#)</a> 
<b>by </b><i>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">B</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></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/>;<br/><a NAME="E3:49_1_1_1_2"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</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="E4:49_1_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T31">TOPMETR:31</a></i>;<br/><a NAME="E5:49_1_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="treal_1.html#K2">(#)</a> </span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="treal_1.html#T11">TREAL_1:11</a></i>;<br/><a NAME="E6:49_1_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></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="borsuk_1.html#T35">BORSUK_1:35</a></i>;<br/><b>then reconsider </b><font color="Maroon">g</font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="treal_1.html#K2">(#)</a> </span>)</span> as   <a href="funct_2.html#NM1">Function</a> of 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>  <b>by </b><i>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a>  ;<br/><a NAME="E9:49_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T7">TOPMETR:7</a></i>;<br/><a NAME="E10:49_1_1_1_2"/>
0 <a href="hidden.html#R1">=</a>  <a href="treal_1.html#K1">(#)</a> 0,1
 <b>by </b><i><a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>;<br/><b>then </b><i><font color="Green">E52</font></i>: <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 = 
 <a href="treal_1.html#K1">(#)</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i>, <a class="ref" href="treal_1.html#D1">TREAL_1:def 1</a></i>
;<br/><a NAME="E12:49_1_1_1_2"/>
1 <a href="hidden.html#R1">=</a> 0,1 <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/><b>then </b><i><font color="Green">E53</font></i>: <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="treal_1.html#K2">(#)</a> 
<b>by </b><i>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i>, <a class="ref" href="treal_1.html#D2">TREAL_1:def 2</a></i>
;<br/><a NAME="E14:49_1_1_1_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="borsuk_2.html#R2" target="_self">are_connected</a> 
 <b>by </b><i>, <a class="txt" href="borsuk_2.html#E1"><i><font color="Green">Lemma30</font></i></a>, </i>;<br/><b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">g</font> as    <a href="borsuk_2.html#M1" target="_self">Path</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font> <b>by </b><i>, <a class="txt" href="borsuk_2.html#E1"><i><font color="Green">Lemma30</font></i></a>, , </i>;<br/><b>take </b><font color="Maroon">h</font> =  <a href="borsuk_2.html#K2" target="_self">-</a> <font color="Maroon">P</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1" target="_self">BORSUK_2:def 1</a><br/><a NAME="E16:49_1_1_1_2"/>
 ex <font color="Olive">t</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<a href="topmetr.html#K5">I[01]</a>  st <br/>( <font color="Olive">t</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Olive">t</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="txt" href="borsuk_2.html#E1"><i><font color="Green">Lemma30</font></i></a>, , </i>;<br/><a NAME="E17:49_1_1_1_2"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="borsuk_2.html#R2" target="_self">are_connected</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E18:49_1_1_1_2"/>
( <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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