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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:7"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> ) )</span><br/>

<a NAME="E3:7"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">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="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></span>;<br/>
<a NAME="E4:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_4.html#T38" title="FDIFF_4:th.38">FDIFF_4:38</a></span>;<br/>
<a NAME="E5:7"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:7_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">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"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T29" title="RFUNCT_1:th.29">RFUNCT_1:29</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E2:7_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></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"><span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="fdiff_4.html#T38" title="FDIFF_4:th.38">FDIFF_4:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T191" title="XCMPLX_1:th.191">XCMPLX_1:191</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:7_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:7"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:7"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:7"/><span class="kw">then </span>
(  <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></span>;<br/>
<a NAME="E9:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A8</font></span>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></span>;<br/>
<a NAME="E10:7"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:7"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">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="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="jordan16.html#D3" title="JORDAN16:def.3">JORDAN16:def 3</a></span>;<br/>
<a NAME="E12:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_4.html#T38" title="FDIFF_4:th.38">FDIFF_4:38</a></span>;<br/>
<a NAME="E13:7"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E14:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></span>;<br/>
<a NAME="E15:7"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:7"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E12:7"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="fdiff_1.html#T28" title="FDIFF_1:th.28">FDIFF_1:28</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></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"><span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="fdiff_4.html#T38" title="FDIFF_4:th.38">FDIFF_4:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T75" title="XCMPLX_1:th.75">XCMPLX_1:75</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:7"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T92" title="XCMPLX_1:th.92">XCMPLX_1:92</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:7_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:7"/>
for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_3"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span><br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> = 
<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"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E10:7"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E9:7"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E14:7"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="fdiff_1.html#T27" title="FDIFF_1:th.27">FDIFF_1:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:7"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:7"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:7"><span class="lab"><font color="Green" title="E23">A14</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</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="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K22" title="SIN_COS:func.22">sin</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos4.html#T19" title="SIN_COS4:th.19">SIN_COS4:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E3:7_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:7"/>
( <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></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">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p0">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p5">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p0">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K47" title="VALUED_1:func.47">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p5">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p4">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="jordan16.html#K1" title="JORDAN16:func.1">AffineMap</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font></span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E10:7"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E9:7"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E14:7"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="fdiff_1.html#T27" title="FDIFF_1:th.27">FDIFF_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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