<?xml version="1.0"?>
<div class="add">

<span class="kw">set </span><font color="Maroon" title="c1">a</font> = 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">b</font> = 2;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">x</font> =  <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">y</font> =  <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Z</font> = <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">C</font> being   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E2:4_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">1,2</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">C</font>
; <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="c6">C</font> is  <a href="integr1c.html#V1" title="INTEGR1C:attr.1">C1-curve-like</a> </span><br/>

<a NAME="E3:4_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p2">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<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="E4:4_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="valued_1.html#K2" title="VALUED_1:func.2">+</a> <span class="p3">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a>  <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">1,2</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>;<br/>
<a NAME="E5:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A6</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">1,2</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:4_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/>
<a NAME="E6:4_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="rcomp_1.html#V3" title="RCOMP_1:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rcomp_1.html#T25" title="RCOMP_1:th.25">RCOMP_1:25</a></span>;<br/>
<a NAME="E7:4_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">1,2</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_1.html#T47" title="XXREAL_1:th.47">XXREAL_1:47</a></span>;<br/>
<a NAME="E8:4_1_1"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T72" title="SIN_COS:th.72">SIN_COS:72</a></span>;<br/>
<a NAME="E9:4_1_1"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="fdiff_1.html#R2" title="FDIFF_1:pred.2">is_differentiable_on</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="fdiff_1.html#T34" title="FDIFF_1:th.34">FDIFF_1:34</a>, <a class="ref" href="sin_cos.html#T73" title="SIN_COS:th.73">SIN_COS:73</a></span>;<br/>
<a NAME="E10:4_1_1"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E12">A12</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></span>
;<br/>
<a NAME="E2:4_1_1_1"/>
for <font color="Olive" title="b1">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  holds <br/><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> <font color="Olive" title="b1">x0</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="partfun1.html#K2" title="PARTFUN1: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">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:4_1_1_1"/>
 <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_1_1_1"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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/>


</div><span class="kw">end;</span></div>
<a NAME="E11:4_1_1"/>
<a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:4_1_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></span>;<br/>
<a NAME="E12:4_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a>, <a class="ref" href="fcont_1.html#T22" title="FCONT_1:th.22">FCONT_1:22</a></span>;<br/>
<a NAME="E13:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A13</font></span>: 
 <a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T62" title="RFUNCT_1:th.62">RFUNCT_1:62</a></span>;<br/>
<span class="lab"><font color="Green" title="E13">A14</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T62" title="RFUNCT_1:th.62">RFUNCT_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a>, <a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E15:4_1_1"/>
for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> ,<font color="Olive" title="b1">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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"> ( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> 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"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> ,<font color="Maroon" title="c7">x1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:4_1_1_3"/><span class="lab"><font color="Green" title="E14">A15</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <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"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> ,<font color="Maroon" title="c7">x1</font></span><br/>

<span class="lab"><font color="Green" title="E15">A16</font></span>: <span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T62" title="RFUNCT_1:th.62">RFUNCT_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_1_1_3"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.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> <font color="Maroon" title="c7">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_1.html#T74" title="RFUNCT_1:th.74">RFUNCT_1:74</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_1_1_3"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/>
 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> ,<font color="Maroon" title="c7">x1</font> = 
 <a href="real_1.html#K1" title="REAL_1:func.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> <font color="Maroon" title="c7">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T72" title="SIN_COS:th.72">SIN_COS:72</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_1_1_3"><span class="lab"><font color="Green" title="E15">A16</font></span></a>, <a class="txt" href="#E1:4_1_1_3"><span class="lab"><font color="Green" title="E14">A15</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:4_1_1_3"/>
<span class="p1">(<span class="default"><a href="valued_1.html#K32" title="VALUED_1:func.32">-</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> ,<font color="Maroon" title="c7">x1</font>
 ; <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="E16:4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A17</font></span>: 
<a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:4_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E13:4_1_1"><span class="lab"><font color="Green" title="E12">A13</font></span></a>, <a class="txt" href="#E14:4_1_1"><span class="lab"><font color="Green" title="E13">A14</font></span></a>, <a class="ref" href="fdiff_1.html#D8" title="FDIFF_1:def.8">FDIFF_1:def 8</a></span>;<br/>
<a NAME="E17:4_1_1"/><span class="lab"><font color="Green" title="E15">A18</font></span>: 
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E16">A19</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
 <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></span>
;<br/>
<a NAME="E2:4_1_1_4"/>
for <font color="Olive" title="b1">x0</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  holds <br/><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> <font color="Olive" title="b1">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1: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">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:4_1_1_4"/>
 <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_1_1_4"><span class="lab"><font color="Green" title="E16">A19</font></span></a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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/>


</div><span class="kw">end;</span></div>
<a NAME="E18:4_1_1"/><span class="lab"><font color="Green" title="E16">A20</font></span>: 
<a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:4_1_1"><span class="lab"><font color="Green" title="E15">A18</font></span></a>, <a class="ref" href="fcont_1.html#T17" title="FCONT_1:th.17">FCONT_1:17</a></span>;<br/>
<span class="lab"><font color="Green" title="E17">A21</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS:27</a></span>
;<br/>
<a NAME="E20:4_1_1"/>
for <font color="Olive" title="b1">x1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> holds <br/><span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> ,<font color="Olive" title="b1">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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"> ( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> implies <span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> ,<font color="Maroon" title="c7">x1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:4_1_1_6"/><span class="lab"><font color="Green" title="E18">A22</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 ; <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="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> ,<font color="Maroon" title="c7">x1</font></span><br/>

 <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> ,<font color="Maroon" title="c7">x1</font> = 
<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> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T73" title="SIN_COS:th.73">SIN_COS:73</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_1_1_6"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:4_1_1_6"/>
<span class="p1">(<span class="default"><a href="sin_cos.html#K23" title="SIN_COS:func.23">cos</a>  <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fdiff_1.html#K1" title="FDIFF_1:func.1">diff</a> <a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a> ,<font color="Maroon" title="c7">x1</font>
 ; <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="E21:4_1_1"/><span class="kw">then </span>
<a href="sin_cos.html#K20" title="SIN_COS:func.20">sin</a>  <a href="fdiff_1.html#K2" title="FDIFF_1:func.2">`|</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 5</span>)</span>,5</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span> is  <a href="fcont_1.html#V1" title="FCONT_1:attr.1">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:4_1_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E18:4_1_1"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="txt" href="#E19:4_1_1"><span class="lab"><font color="Green" title="E17">A21</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="kw">hence </span><a NAME="E22:4_1_1"/>
<font color="Maroon" title="c6">C</font> is  <a href="integr1c.html#V1" title="INTEGR1C:attr.1">C1-curve-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E5:4_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E6:4_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="txt" href="#E7:4_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E8:4_1_1"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E9:4_1_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E16:4_1_1"><span class="lab"><font color="Green" title="E14">A17</font></span></a>, <a class="ref" href="integr1c.html#D3" target="_self" title="INTEGR1C:def.3">Def3</a>, <a class="ref" href="sin_cos.html#T27" title="SIN_COS:th.27">SIN_COS: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/>


</div>
