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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">m</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span><br/>





<a NAME="E1:23"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E2:23"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#D5" onmouseover="rs('valued_1/D5')" onmouseout="rh()">VALUED_1:def 5</a></span>;<br/>
<a NAME="E3:23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E4:23"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">F</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="23_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">F</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:23_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">F</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/>

<a NAME="E2:23_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E2:23_1"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T7" target="_self" onmouseover="rs('euler_2/T7')" onmouseout="rh()">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> <font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T6" onmouseover="rs('valued_1/T6')" onmouseout="rh()">VALUED_1:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E1:23_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:23_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>
<a NAME="E5:23"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:23"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E6:23"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="euler_2.html#D1" target="_self" onmouseover="rs('euler_2/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:23"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="valued_1.html#K24" title="VALUED_1:func.24">(#)</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="euler_2.html#K2" title="EULER_2:func.2">mod</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:23"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E4:23"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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