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<b>let </b><font color="Maroon">f</font> be  <a href="funct_1.html#V2">one-to-one</a> <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<a href="topmetr.html#K3">R^1</a> ;<br/>

<a NAME="E1:14"/><i><font color="Green">E28</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#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T35">BORSUK_1:35</a>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<i><font color="Green">E38</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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> = 
<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="subset_1.html#K8">/\</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<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="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_2"/>
( <font color="Maroon">g</font> <a href="rfunct_2.html#R1">is_increasing_on</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> or <font color="Maroon">g</font> <a href="rfunct_2.html#R2">is_decreasing_on</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> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_1"><b>suppose </b></a><a NAME="E1:14_2_1"/>
<font color="Maroon">g</font> <a href="rfunct_2.html#R1">is_increasing_on</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_2_1"/>
( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">fx</font>, <font color="Olive">fy</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> &amp; <font color="Olive">fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">fx</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">fy</font> or for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">fx</font>, <font color="Olive">fy</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> &amp; <font color="Olive">fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">fx</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">fy</font> )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rfunct_2.html#D2">RFUNCT_2:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_2_2"><b>suppose </b></a><a NAME="E1:14_2_2"/>
<font color="Maroon">g</font> <a href="rfunct_2.html#R2">is_decreasing_on</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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_2_2"/>
( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">fx</font>, <font color="Olive">fy</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> &amp; <font color="Olive">fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">fx</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">fy</font> or for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font>, <font color="Olive">fx</font>, <font color="Olive">fy</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">q</font> &amp; <font color="Olive">fx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> &amp; <font color="Olive">fy</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> holds <br/><font color="Olive">fx</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">fy</font> )
 <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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