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<span class="kw">let </span><font color="Maroon" title="c5">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">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> holds <br/> ( <font color="Maroon" title="c5">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4 iff <font color="Maroon" title="c5">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</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="c5">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4 iff <font color="Maroon" title="c5">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> )</span><br/>



<span class="kw">thus </span><a NAME="E1:164"/>
( <font color="Maroon" title="c5">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4 implies <font color="Maroon" title="c5">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</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"> ( <font color="Maroon" title="c5">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span> implies <font color="Maroon" title="c5">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4 )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="164_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:164_1"/>
<font color="Maroon" title="c5">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4
 ; <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="c5">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">r</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:164_1"/><span class="lab"><font color="Green" title="E124">A1</font></span>: 
<font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">and </span><br/><a NAME="E4:164_1"/><span class="lab"><font color="Green" title="E125">A2</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T2" onmouseover="rs('groupp_1/T2')" onmouseout="rh()">GROUPP_1:2</a></span>;<br/>
<a NAME="E5:164_1"/>
 not  not <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:164_1"><span class="lab"><font color="Green" title="E125">A2</font></span></a></span>;<br/>
<a NAME="E6:164_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:164_1"><span class="lab"><font color="Green" title="E124">A1</font></span></a>, <a class="ref" href="newton.html#T4" onmouseover="rs('newton/T4')" onmouseout="rh()">NEWTON:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:164_1"/>
<font color="Maroon" title="c5">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</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><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:164"/>
<font color="Maroon" title="c5">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="enumset1.html#K3" title="ENUMSET1:func.3">{</a><span class="default">1,<font color="Maroon" title="c6">p</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span>)</span></span><a href="enumset1.html#K3" title="ENUMSET1:func.3">}</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"> <font color="Maroon" title="c5">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4</span><br/>

<a NAME="E3:164"/><span class="kw">then </span>
( <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 1 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 2 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 3 or <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D3" onmouseover="rs('enumset1/D3')" onmouseout="rh()">ENUMSET1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:164"/>
<font color="Maroon" title="c5">m</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> 4
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T12" onmouseover="rs('int_2/T12')" onmouseout="rh()">INT_2:12</a>, <a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</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>
