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<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><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="E1:36"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">f</font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font> &amp; ( for <font color="Olive">x0</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x0</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font> 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">x0</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_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="E1:36_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font></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="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font></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="E2:36_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font> &amp; <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="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</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">rr</font> =  <a href="square_1.html#K1">min</a> <font color="Maroon">r1</font>,<font color="Maroon">r2</font>;<br/><a NAME="E3:36_1"/><i><font color="Green">E46</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>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span> <a href="tarski.html#R1">c=</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="limfunc1.html#T16">LIMFUNC1:16</a></i>;<br/><a NAME="E4:36_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">g</font> where <font color="Olive">g</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/><a NAME="E5:36_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 ex <font color="Olive">g1</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> &amp; <font color="Olive">g1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 ;<br/><a NAME="E6:36_1"/>
<font color="Maroon">r2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="prob_1.html#D15">PROB_1:def 15</a></i>;<br/><a NAME="E7:36_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 ex <font color="Olive">g2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> &amp; <font color="Olive">g2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 ;<br/><a NAME="E8:36_1"/><i><font color="Green">E50</font></i>: 
(  <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="E9:36_1"/><b>then </b><i><font color="Green">E51</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
 <b>by </b><i><a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><a NAME="E10:36_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> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:36_1"/><b>then </b><i><font color="Green">E52</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>,<font color="Maroon">r</font></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="E12:36_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> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E13:36_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>,<font color="Maroon">r</font></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="E14:36_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>,<font color="Maroon">r</font></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>, </i>;<br/><a NAME="E15:36_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_2.html#R3">is_non_decreasing_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>,<font color="Maroon">r</font></span><a href="rcomp_1.html#K2">.[</a></span>
 <b>by </b><i>, , <a class="ref" href="rolle.html#T11">ROLLE:11</a></i>;<br/><a NAME="E16:36_1"/>
<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="E17:36_1"/><b>then </b>
<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> <font color="Maroon">r</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E18:36_1"/><b>then </b>
<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>,<font color="Maroon">r</font></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="E19:36_1"/><b>then </b><i><font color="Green">E65</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>,<font color="Maroon">r</font></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="E20:36_1"/>
<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>,<font color="Maroon">r</font></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="E21:36_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#D4">RFUNCT_2:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:36"/>
<font color="Maroon">f</font> <a href="rfunct_2.html#R3">is_non_decreasing_on</a>  <a href="limfunc1.html#NK2" target="_self">left_open_halfline</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#D4">RFUNCT_2:def 4</a></i>;<br/>


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