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<div><span class="kw">theorem </span><a NAME="T37"><span class="comment"><font color="firebrick">:: DIFF_1:37</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">h</font>, <font color="Olive" title="b2">x</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">f1</font>, <font color="Olive" title="b4">f2</font> being   <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/>  for <font color="Olive" title="b5">S</font> being   <a href="diff_1.html#NM1" title="DIFF_1:NM.1">Seq_Sequence</a>  st (  for <font color="Olive" title="b6">n</font>, <font color="Olive" title="b7">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">n</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b5">S</font> <a href="diff_1.html#K12" title="DIFF_1:func.12">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b6">n</font> <a href="newton.html#K4" title="NEWTON:func.4">choose</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="diff_1.html#NK7" title="DIFF_1:NK.7">fdif</a> (<font color="Olive" title="b3">f1</font>,<font color="Olive" title="b1">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b7">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0: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#NK7" title="DIFF_1:NK.7">fdif</a> (<font color="Olive" title="b4">f2</font>,<font color="Olive" title="b1">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b7">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">x</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">h</font></span>)</span></span>)</span></span>)</span> ) holds <br/>( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK7" title="DIFF_1:NK.7">fdif</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b4">f2</font></span>)</span>,<font color="Olive" title="b1">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K6" title="SERIES_1:func.6">Sum</a> (<span class="p1">(<span class="default"><font color="Olive" title="b5">S</font> <a href="diff_1.html#K12" title="DIFF_1:func.12">.</a> 1</span>)</span>,1) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="diff_1.html#NK7" title="DIFF_1:NK.7">fdif</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">f1</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b4">f2</font></span>)</span>,<font color="Olive" title="b1">h</font>)</span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> 2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="series_1.html#K6" title="SERIES_1:func.6">Sum</a> (<span class="p1">(<span class="default"><font color="Olive" title="b5">S</font> <a href="diff_1.html#K12" title="DIFF_1:func.12">.</a> 2</span>)</span>,2) )</div></div>
