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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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>let </b><font color="Maroon">h</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f</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">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/>













<b>assume </b><a NAME="E1:29"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="partfun3.html#K9">(#)</a> <font color="Maroon">f1</font></span>)</span> <a href="partfun3.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="partfun3.html#K9">(#)</a> <font color="Maroon">f2</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">g1</font> = <font color="Maroon">r1</font> <a href="partfun3.html#K9">(#)</a> <font color="Maroon">f1</font>;<br/>
<b>set </b><font color="Maroon">g2</font> = <font color="Maroon">r2</font> <a href="partfun3.html#K9">(#)</a> <font color="Maroon">f2</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" target="_self">bdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <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 class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="diff_1.html#NK12" target="_self">bdif</a> <span class="p4">(<span class="default"><font color="Maroon">r1</font> <a href="partfun3.html#K9">(#)</a> <font color="Maroon">f1</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <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> <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#NK12" target="_self">bdif</a> <span class="p4">(<span class="default"><font color="Maroon">r2</font> <a href="partfun3.html#K9">(#)</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <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>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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#NK12" target="_self">bdif</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <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#NK12" target="_self">bdif</a> <span class="p4">(<span class="default"><font color="Maroon">r2</font> <a href="partfun3.html#K9">(#)</a> <font color="Maroon">f2</font></span>)</span>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <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>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <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#NK12" target="_self">bdif</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <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"><font color="Maroon">r2</font> <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#NK12" target="_self">bdif</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"><font color="Maroon">n</font> <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>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E3:29"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" target="_self">bdif</a> <font color="Maroon">f</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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"><font color="Maroon">r1</font> <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#NK12" target="_self">bdif</a> <font color="Maroon">f1</font>,<font color="Maroon">h</font></span>)</span> <a href="seqfunc.html#K1">.</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <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"><font color="Maroon">r2</font> <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#NK12" target="_self">bdif</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"><font color="Maroon">n</font> <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>
 ;<br/>


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