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<span class="kw">let </span><font color="Maroon" title="c1">i</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">zero</a>  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">i</font> <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> </span><br/>

<a NAME="E1:54"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:54"/><span class="lab"><font color="Green" title="E32">a1</font></span>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> or <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D1" title="INT_1:def.1">INT_1:def 1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_1"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> or <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E32">a1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><span class="kw">suppose </span></a><a NAME="E1:54_1_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">i</font> <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> </span><br/><div class="add"><a NAME="E2:54_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><span class="kw">hence </span><a NAME="E3:54_1_1"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">i</font> <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="funct_9.html#E46"><span class="lab"><font color="Green" title="E31">Lmcosm</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><span class="kw">suppose </span></a><a NAME="E1:54_1_2"/><span class="lab"><font color="Green" title="E33">a2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">i</font> <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:54_1_2"/><span class="lab"><font color="Green" title="E34">a3</font></span>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></span>;<br/><a NAME="E4:54_1_2"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="funct_9.html#E46"><span class="lab"><font color="Green" title="E31">Lmcosm</font></span></a></span>;<br/><a NAME="E5:54_1_2"/><span class="kw">then </span>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_9.html#T14" target="_self" title="FUNCT_9:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E6:54_1_2"/>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><a href="sin_cos.html#K20" title="SIN_COS:func.20">cos</a></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> is <a href="sin_cos.html#K33" title="SIN_COS:func.33">PI</a> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">i</font> <a href="funct_9.html#V1" title="FUNCT_9:attr.1">-periodic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_2"><span class="lab"><font color="Green" title="E33">a2</font></span></a>, <a class="txt" href="#E3:54_1_2"><span class="lab"><font color="Green" title="E34">a3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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