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<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <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">  <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">F0</font> =  <a href="number08.html#K1" title="NUMBER08:func.1">Euler_factorization</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">F1</font> =  <a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">F2</font> =  <a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">b0</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="number08.html#K1" title="NUMBER08:func.1">Euler_factorization</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">b1</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">b2</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>
<a NAME="E1:74"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K1" title="NUMBER08:func.1">Euler_factorization</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T26" target="_self" onmouseover="rs('number08/T26')" onmouseout="rh()">Th26</a></span>;<br/>
<a NAME="E2:74"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T33" target="_self" onmouseover="rs('number08/T33')" onmouseout="rh()">Th33</a></span>;<br/>
<a NAME="E3:74"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T45" target="_self" onmouseover="rs('number08/T45')" onmouseout="rh()">Th45</a>, <a class="ref" href="number08.html#T44" target="_self" onmouseover="rs('number08/T44')" onmouseout="rh()">Th44</a></span>;<br/>
<a NAME="E4:74"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T12" target="_self" onmouseover="rs('number08/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E5:74"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
 <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</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> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T28" target="_self" onmouseover="rs('number08/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E6:74"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T12" target="_self" onmouseover="rs('number08/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E7:74"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
 <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</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> <span class="p1">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T30" target="_self" onmouseover="rs('number08/T30')" onmouseout="rh()">Th30</a></span>;<br/>
<a NAME="E8:74"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#D2" target="_self" onmouseover="rs('number08/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E9:74"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#D3" target="_self" onmouseover="rs('number08/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E10:74"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="number08.html#K1" title="NUMBER08:func.1">Euler_factorization</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="valued_1.html#K18" title="VALUED_1:func.18">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T31" target="_self" onmouseover="rs('number08/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E11:74"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A10</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K1" title="NUMBER08:func.1">Euler_factorization</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:74"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E5:74"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E6:74"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E7:74"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E8:74"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E9:74"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="ref" href="number08.html#T16" target="_self" onmouseover="rs('number08/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<a NAME="E12:74"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a>  <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="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="newton.html#K7" title="NEWTON:func.7">!</a></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T15" onmouseover="rs('newton/T15')" onmouseout="rh()">NEWTON:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:74"/>
 <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">n</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font> <a href="newton.html#K7" title="NEWTON:func.7">!</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:74"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:74"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:74"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E11:74"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="newton02.html#T2" onmouseover="rs('newton02/T2')" onmouseout="rh()">NEWTON02:2</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/>


</div>
