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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E7">Th4</font></span>: <a NAME="T4"><span class="comment"><font color="firebrick">:: CURVE:4</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><br/>  for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">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>  st <font color="Olive" title="b1">a</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b3">x</font> is  <a href="fdiff_1.html#V4" title="FDIFF_1:attr.4">differentiable</a>  &amp; <font color="Olive" title="b4">y</font> is  <a href="fdiff_1.html#V4" title="FDIFF_1:attr.4">differentiable</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <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="b3">x</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <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="b4">y</font> &amp; <font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font></span>)</span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font></span>)</span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; (  for <font color="Olive" title="b5">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font></span>)</span> holds <br/> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</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="b3">x</font>,<font color="Olive" title="b5">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"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b4">y</font>,<font color="Olive" title="b5">t</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> ) holds <br/> ex <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ex <font color="Olive" title="b7">L</font> being   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  <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> st <br/>( <font color="Olive" title="b5">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">b1</font> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b7">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; (  for <font color="Olive" title="b8">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b8">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"><font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">t</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="taylor_1.html#K4" title="TAYLOR_1:func.4">#R</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p5">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p5">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span></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="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p5">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p4">(<span class="default"><font color="Olive" title="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p5">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b5">a1</font>,<font color="Olive" title="b8">t</font>) ) &amp; (  for <font color="Olive" title="b8">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> holds <br/><span class="p1">(<span class="default"><a href="curve.html#K1" title="CURVE:func.1">ArcLenP</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">t</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span> ) &amp; <font color="Olive" title="b7">L</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  &amp; <font color="Olive" title="b7">L</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <font color="Olive" title="b7">L</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> &amp; (  for <font color="Olive" title="b8">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b8">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"><font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><font color="Olive" title="b7">L</font> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b8">t</font> ) &amp; <font color="Olive" title="b7">L</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"><font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font></span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> &amp; (  for <font color="Olive" title="b8">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b8">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"><font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font></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="b7">L</font>,<font color="Olive" title="b8">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b8">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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b4">y</font>,<font color="Olive" title="b8">t</font>)</span>)</span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="prepower.html#K6" title="PREPOWER:func.6">#R</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2</span>)</span> ) &amp; <font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> &amp; (  for <font color="Olive" title="b8">t</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b8">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> holds <br/> <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span>,<font color="Olive" title="b8">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b7">L</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">t</font></span>)</span>)</span>)</span> ) &amp; <font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a>  is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; (  for <font color="Olive" title="b8">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b7">L</font> holds <br/>( <font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b8">s</font> &amp; <font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="fdiff_1.html#R1" title="FDIFF_1:pred.1">is_differentiable_in</a> <font color="Olive" title="b8">s</font> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span>,<font color="Olive" title="b8">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b3">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">s</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span>,<font color="Olive" title="b8">s</font>)</span>)</span> &amp;  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span>,<font color="Olive" title="b8">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<font color="Olive" title="b4">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">s</font></span>)</span>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span>,<font color="Olive" title="b8">s</font>)</span>)</span> &amp; <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> (<span class="p3">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span>,<font color="Olive" title="b8">s</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"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p3">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span>,<font color="Olive" title="b8">s</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; <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></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="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></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="b4">y</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b4">y</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></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="b7">L</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b7">L</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="rfunct_1.html#K6" title="RFUNCT_1:func.6">^</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> &amp; <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b7">L</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> &amp; <font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> is  <a href="fdiff_1.html#V4" title="FDIFF_1:attr.4">differentiable</a>  &amp; <font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span> is  <a href="fdiff_1.html#V4" title="FDIFF_1:attr.4">differentiable</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span></span>)</span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span></span>)</span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a>  &amp; (  for <font color="Olive" title="b8">s</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>  st <font color="Olive" title="b8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span></span>)</span> holds <br/><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> (<span class="p3">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span>,<font color="Olive" title="b8">s</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"><a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> (<span class="p3">(<span class="default"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span>,<font color="Olive" title="b8">s</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;  <a href="integra5.html#K4" title="INTEGRA5:func.4">integral</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">y</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">x</font> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b3">x</font></span>)</span></span>)</span></span>)</span>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</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"><font color="Olive" title="b4">y</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p3">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">x</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><font color="Olive" title="b7">L</font> <a href="partfun2.html#K2" title="PARTFUN2:func.2">"</a></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span>)</span>) )</div></div>
