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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E32">Th20</font></span>: <a NAME="T14"><span class="comment"><font color="firebrick">:: PDLAX:14</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</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="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">x</font> is  <a href="fdiff_1.html#V4" title="FDIFF_1:attr.4">differentiable</a>  &amp; <font color="Olive" title="b2">y</font> is  <a href="fdiff_1.html#V4" title="FDIFF_1:attr.4">differentiable</a>  &amp; <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">Z</font> &amp; <font color="Olive" title="b3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; (  for <font color="Olive" title="b4">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b4">t</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"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">t</font></span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp; <font color="Olive" title="b2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Olive" title="b2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> ex <font color="Olive" title="b4">u</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ex <font color="Olive" title="b5">F</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> st <br/>( <font color="Olive" title="b4">u</font> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; <font color="Olive" title="b4">u</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; <font color="Olive" title="b4">u</font> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b2">y</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">u</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="integra5.html#K3" title="INTEGRA5:func.3">['</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="integra5.html#K3" title="INTEGRA5:func.3">']</a></span> &amp; (  for <font color="Olive" title="b6">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b2">y</font>,<font color="Olive" title="b6">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b4">u</font>,<font color="Olive" title="b6">t</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b4">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">t</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K18" title="SIN_COS:func.18">cos</a> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">t</font></span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b6">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Olive" title="b4">u</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p0"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p5">(<span class="default"><font color="Olive" title="b4">u</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p0"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <a href="sin_cos.html#K16" title="SIN_COS:func.16">sin</a></span>)</span></span>)</span>,<span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><font color="Olive" title="b6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p3">(<span class="default"><font color="Olive" title="b6">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>) ) &amp; <font color="Olive" title="b5">F</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>  &amp;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span></span>)</span> &amp;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>) &amp;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="valued_1.html#K3" title="VALUED_1:func.3">+</a> <span class="p3">(<span class="default"><a href="fcont_1.html#K1" title="FCONT_1:func.1">AffineMap</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <font color="Olive" title="b3">Z</font></span>)</span></span>)</span></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="sin_cos.html#K28" title="SIN_COS:func.28">PI</a>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K1" title="SEQ_2:func.1">lim</a> <font color="Olive" title="b5">F</font> )</div></div>
