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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">ps</font>, <font color="Maroon" title="c2">pt</font>, <font color="Maroon" title="c3">f</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</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="c1">ps</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Maroon" title="c2">pt</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Maroon" title="c3">f</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">ps</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">pt</font> &amp;  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</font> implies <font color="Maroon" title="c3">f</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:5"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c1">ps</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</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"> (  not <font color="Maroon" title="c2">pt</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  or  not <font color="Maroon" title="c3">f</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">ps</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">pt</font> or  not  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</font> or <font color="Maroon" title="c3">f</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:5"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c2">pt</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</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"> (  not <font color="Maroon" title="c3">f</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">ps</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">pt</font> or  not  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</font> or <font color="Maroon" title="c3">f</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:5"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">ps</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <font color="Maroon" title="c2">pt</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"> (  not  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</font> or <font color="Maroon" title="c3">f</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E4:5"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</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"> <font color="Maroon" title="c3">f</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> </span><br/>

<a NAME="E5:5"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:5"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="pre_poly.html#T38" onmouseover="rs('pre_poly/T38')" onmouseout="rh()">PRE_POLY:38</a></span>;<br/>
<a NAME="E6:5"/>
 for <font color="Olive" title="b1">x</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c3">f</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c3">f</font> <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> <font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">n</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="5_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="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="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c3">f</font> implies  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:5_1"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</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">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:5_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</font> or <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E5:5"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_1"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c1">ps</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">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> )</span><br/><div class="add"><a NAME="E2:5_1_1_1"/>
<font color="Maroon" title="c1">ps</font> <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="c3">f</font> <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="#E3:5"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:5_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="group_17.html#T3" target="_self" onmouseover="rs('group_17/T3')" onmouseout="rh()">Th3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:5_1_1_1"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_1"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="txt" href="#E1:5"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:5_1_1_2"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c2">pt</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">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> )</span><br/><div class="add"><a NAME="E2:5_1_1_2"/>
<font color="Maroon" title="c2">pt</font> <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="c3">f</font> <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="#E3:5"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="txt" href="#E4:5"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E1:5_1_1_2"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="group_17.html#T4" target="_self" onmouseover="rs('group_17/T4')" onmouseout="rh()">Th4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:5_1_1_2"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:5_1_1_2"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="txt" href="#E2:5"><span class="lab"><font color="Green" title="E7">A2</font></span></a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:5"/>
<font color="Maroon" title="c3">f</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</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>
