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<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</font> being   <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/> for <font color="Olive" title="b3">Z</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">m</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Olive" title="b1">n</font></span>)</span>,<font color="Olive" title="b3">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</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="b1">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">m</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Olive" title="b1">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</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="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">Z</font> be  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</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="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">m</font> &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</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="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:38"/><i><font color="Green" title="E24">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:38"/><i><font color="Green" title="E25">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="txt" href="hfdiff_1.html#E2:38"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a></i>;<br/>
<i><font color="Green" title="E26">A5</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</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"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c4">Z</font></span>)</span> = 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Z</font>
<b>by </b><i><a class="txt" href="hfdiff_1.html#E3:38"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c4">Z</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="hfdiff_1.html#T32" target="_self" title="HFDIFF_1:th.32">Th23</a>, <a class="txt" href="hfdiff_1.html#E1:38"><i><font color="Green" title="E23">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="hfdiff_1.html#E4:38"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="hfdiff_1.html#E1:38"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</font></span>)</span>
<b>by </b><i><a class="ref" href="valued_1.html#D5" title="VALUED_1:def.5">VALUED_1:def 5</a>, <a class="txt" href="hfdiff_1.html#E3:38"><i><font color="Green" title="E25">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="taylor_1.html#D1" title="TAYLOR_1:def.1">TAYLOR_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:38"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="taylor_1.html#K5" title="TAYLOR_1:func.5">diff</a> <span class="p3">(<span class="default"><a href="taylor_1.html#K1" title="TAYLOR_1:func.1">#Z</a> <font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c4">Z</font></span>)</span> <a href="seqfunc.html#K1" title="SEQFUNC:func.1">.</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c1">x</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="Maroon" title="c2">n</font> <a href="newton.html#K9" title="NEWTON:func.9">choose</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="sin_cos.html#K5" title="SIN_COS:func.5">!</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="prepower.html#K5" title="PREPOWER:func.5">#Z</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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