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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">r1</font>, <font color="Maroon" title="c3">r2</font>, <font color="Maroon" title="c4">h</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">f1</font>, <font color="Maroon" title="c7">f2</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="E1:29"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="partfun3.html#K3" title="PARTFUN3:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r2</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c7">f2</font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c2">r1</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c6">f1</font>;<br/>
<b>set </b><font color="Maroon" title="c9">g2</font> = <font color="Maroon" title="c3">r2</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c7">f2</font>;<br/>
<b>let </b><font color="Maroon" title="c10">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">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" title="DIFF_1:NK.12">bdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">r1</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c6">f1</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="DIFF_1:NK.12">bdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r2</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c7">f2</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="txt" href="diff_1.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="diff_1.html#T15" target="_self" title="DIFF_1:th.15">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</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"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</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" title="DIFF_1:NK.12">bdif</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">r2</font> <a href="partfun3.html#K9" title="PARTFUN3:func.9">(#)</a> <font color="Maroon" title="c7">f2</font></span>)</span>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T14" target="_self" title="DIFF_1:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</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"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r2</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"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="diff_1.html#T14" target="_self" title="DIFF_1:th.14">Th14</a></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" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r1</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"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r2</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"><span class="p4">(<span class="default"><a href="diff_1.html#NK12" title="DIFF_1:NK.12">bdif</a> <font color="Maroon" title="c7">f2</font>,<font color="Maroon" title="c4">h</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span>)</span>
 ;<br/>


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