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<b>let </b><font color="Maroon">h</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>







<a NAME="E1:59"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E2:59"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:59"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</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">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:59"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<a NAME="E5:59"/>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
;<br/>
<a NAME="E6:59"/><b>then </b>
( <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:59"/><b>then </b>
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</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">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E8:59"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="diff_1.html#K3" target="_self">fD</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <span class="p4">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="diff_1.html#K4" target="_self">fD</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D5">SEQ_1:def 5</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K4" target="_self">fD</a> <font color="Maroon">f2</font>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K4" target="_self">fD</a> <font color="Maroon">f2</font>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K4" target="_self">fD</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K3" target="_self">fD</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f2</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K4" target="_self">fD</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K3" target="_self">fD</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f2</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K3" target="_self">fD</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f2</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#K3" target="_self">fD</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 0</span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f2</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E10:59"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="partfun3.html#K5">(#)</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f2</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK10" target="_self">fdif</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> 1</span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">f2</font> <a href="seq_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="real_1.html#K3">+</a> <font color="Maroon">h</font></span>)</span></span>)</span></span>)</span>
 ;<br/>


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