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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="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">f</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">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D4" target="_self" onmouseover="rs('number08/D4')" onmouseout="rh()">NUMBER08:def 4</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> <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> implies <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:72"/><span class="lab"><font color="Green" title="E57">A1</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> <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 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"><a href="number08.html#K3" title="NUMBER08:func.3">Euler_factorization_2</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">x</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="number08.html#T29" target="_self" onmouseover="rs('number08/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<a NAME="E3:72"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="number08.html#D3" target="_self" onmouseover="rs('number08/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<a NAME="E4:72"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
 <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> <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#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<a NAME="E5:72"/>
 <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="E6:72"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="nat_3.html#T36" onmouseover="rs('nat_3/T36')" onmouseout="rh()">NAT_3:36</a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:72"/>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</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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