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<b>let </b><font color="Maroon" title="c1">x0</font>, <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>







<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x1</font>
 ;<br/>

<a NAME="E2:48"/>
( <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
;<br/>
<a NAME="E3:48"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> &amp; <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E1:48"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="diff_1.html#D9" target="_self" title="DIFF_1:def.9">Def9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E3:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">x1</font></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E3:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="seq_1.html#D6" title="SEQ_1:def.6">SEQ_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">x1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c1">x0</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">x1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></i>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span>
<b>by </b><i><a class="txt" href="diff_1.html#E1:48"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="diff_1.html#D9" target="_self" title="DIFF_1:def.9">Def9</a></i>
;<br/>
<b>hence </b><a NAME="E5:48"/>
<span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c4">f</font></span>)</span>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="diff_1.html#K12" title="DIFF_1:func.12">[!</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c1">x0</font>,<font color="Maroon" title="c2">x1</font></span><a href="diff_1.html#K12" title="DIFF_1:func.12">!]</a></span>
 ;<br/>


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