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

<b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  holds 0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">x0</font> ) )
 ;<br/>

<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> </span>)</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> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> </span>)</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> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 ;<br/><a NAME="E2:42_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>set </b><font color="Maroon">rn</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>;<br/><b>set </b><font color="Maroon">rx</font> =  <a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>;<br/><a NAME="E3:42_1"/>
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/><a NAME="E4:42_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/><a NAME="E5:42_1"/>
(  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp;  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/><a NAME="E6:42_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> <a href="real_1.html#K5">-</a> 0 &amp; <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> 0 )
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E7:42_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E8:42_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1 &amp; <font color="Maroon">r2</font> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E9:42_1"/><b>then </b>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g1</font> where <font color="Olive">g1</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E10:42_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E11:42_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g2</font> where <font color="Olive">g2</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">g2</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1 ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E12:42_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D2">RCOMP_1:def 2</a></i>;<br/><a NAME="E13:42_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:42_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="fdiff_1.html#T34">FDIFF_1:34</a></i>;<br/><a NAME="E15:42_1"/>
for <font color="Olive">g1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span> holds <br/>0 <a href="xxreal_0.html#R1">&lt;</a>  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">f</font>,<font color="Olive">g1</font>
 <b>by </b><i/>;<br/><a NAME="E16:42_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="rolle.html#T9">ROLLE:9</a></i>;<br/><a NAME="E17:42_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</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>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E18:42_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K2">max</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font></span>)</span> <a href="real_1.html#K3">+</a> 1</span>)</span></span><a href="rcomp_1.html#K2">.[</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>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E19:42_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font>
 <b>by </b><i>, , , <a class="ref" href="rfunct_2.html#D2">RFUNCT_2:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:42"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#R1">is_increasing_on</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="rfunct_2.html#D2">RFUNCT_2:def 2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><a NAME="E1:42_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">r2</font> )
 <b>and </b><br/><a NAME="E2:42_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r2</font>
 ;<br/><a NAME="E3:42_2"/>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  )
 ;<br/><a NAME="E4:42_2"/><b>then </b>
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/><a NAME="E5:42_2"/><b>then </b><i><font color="Green">E68</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> </span>)</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> &amp; <font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> </span>)</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> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_2_1_1"/>
( <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font> or <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1_1"><b>suppose </b></a><a NAME="E1:42_2_1_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_2_1_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_2_1_1_2"><b>suppose </b></a><a NAME="E1:42_2_1_1_2"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_2_1_1_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:42_2"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:42"/>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun1.html#T38">PARTFUN1:38</a></i>;<br/>


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