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<span class="kw">let </span><font color="Maroon" title="c1">h</font>, <font color="Maroon" title="c2">k</font>, <font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</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"> ( <font color="Maroon" title="c1">h</font>,<font color="Maroon" title="c2">k</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">k</font> &amp; (  for <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> holds <br/> not <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">h</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) &amp; (  for <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> holds <br/> not <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c2">k</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) implies ( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:8"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c1">h</font>,<font color="Maroon" title="c2">k</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</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">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">k</font> or  ex <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> &amp; <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">h</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) or  ex <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> &amp; <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c2">k</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) or ( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:8"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">k</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">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> &amp; <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">h</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) or  ex <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> &amp; <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c2">k</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) or ( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:8"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
 for <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> holds <br/> not <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c1">h</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</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"> (  ex <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> st <br/>( <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> &amp; <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c2">k</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  ) or ( <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:8"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 for <font color="Olive" title="b1">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">q</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> holds <br/> not <font color="Olive" title="b1">q</font>,<font color="Maroon" title="c2">k</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</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="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">h</font> &amp; <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">ps</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">ph</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">pt</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">pk</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font>;<br/>
<a NAME="E5:8"/><span class="lab"><font color="Green" title="E13">A5</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="group_17.html#T6" target="_self" onmouseover="rs('group_17/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E6:8"/><span class="lab"><font color="Green" title="E14">A6</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="group_17.html#T6" target="_self" onmouseover="rs('group_17/T6')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E7:8"/>
 <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="c1">h</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T44" onmouseover="rs('nat_3/T44')" onmouseout="rh()">NAT_3:44</a>, <a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/>
<a NAME="E8:8"/><span class="kw">then </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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">k</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="E9:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> <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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</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/>
<span class="kw">set </span><font color="Maroon" title="c9">f</font> = <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">g</font> = <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span>;<br/>
<a NAME="E10:8"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> 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="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="xboole_1.html#T64" onmouseover="rs('xboole_1/T64')" onmouseout="rh()">XBOOLE_1:64</a>, <a class="txt" href="group_17.html#E5"><span class="lab"><font color="Green" title="E5">Lm1</font></span></a></span>;<br/>
<a NAME="E11:8"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span> 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="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="group_17.html#E5"><span class="lab"><font color="Green" title="E5">Lm1</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E18">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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span> = 
<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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="nat_3.html#T19" onmouseover="rs('nat_3/T19')" onmouseout="rh()">NAT_3:19</a></span>
<br/>.= 
<font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
<br/>.= 
<font color="Maroon" title="c1">h</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
;<br/>
<span class="lab"><font color="Green" title="E19">A11</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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span> = 
<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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="xboole_1.html#T64" onmouseover="rs('xboole_1/T64')" onmouseout="rh()">XBOOLE_1:64</a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="nat_3.html#T19" onmouseover="rs('nat_3/T19')" onmouseout="rh()">NAT_3:19</a></span>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
<br/>.= 
<font color="Maroon" title="c3">s</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
;<br/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span></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> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T38" onmouseover="rs('pre_poly/T38')" onmouseout="rh()">PRE_POLY:38</a></span>
<br/>.= 
 <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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="int_7.html#T15" onmouseover="rs('int_7/T15')" onmouseout="rh()">INT_7:15</a>, <a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E12:8"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span></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> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T38" onmouseover="rs('pre_poly/T38')" onmouseout="rh()">PRE_POLY:38</a></span>
;<br/>
<a NAME="E15:8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A12</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> &amp;  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="group_17.html#T1" target="_self" onmouseover="rs('group_17/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E16:8"/><span class="lab"><font color="Green" title="E21">A13</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</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="c1">h</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<a NAME="E17:8"/><span class="lab"><font color="Green" title="E22">A14</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#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</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">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<a NAME="E18:8"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">h</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">h</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</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"> ( <font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">h</font></span>)</span> implies <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">h</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_4"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c1">h</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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">h</font></span>)</span></span><br/>

<a NAME="E2:8_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</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="E3:8_4"/><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_4"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E15:8"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T3" target="_self" onmouseover="rs('group_17/T3')" onmouseout="rh()">Th3</a>, <a class="txt" href="#E3:8_4"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="xboole_1.html#T64" onmouseover="rs('xboole_1/T64')" onmouseout="rh()">XBOOLE_1:64</a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T3" target="_self" onmouseover="rs('group_17/T3')" onmouseout="rh()">Th3</a>, <a class="txt" href="#E2:8_4"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="int_7.html#T15" onmouseover="rs('int_7/T15')" onmouseout="rh()">INT_7:15</a>, <a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E12:8"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E19">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_4"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</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>
<a NAME="E19:8"/><span class="lab"><font color="Green" title="E24">A19</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">k</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">p</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"> ( <font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">k</font></span>)</span> implies <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">k</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:8_5"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
<font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">k</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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">k</font></span>)</span></span><br/>

<a NAME="E2:8_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A21</font></span>: 
<font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</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="E3:8_5"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
<font color="Maroon" title="c11">p</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> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:8"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E1:8_5"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T4" target="_self" onmouseover="rs('group_17/T4')" onmouseout="rh()">Th4</a>, <a class="txt" href="#E3:8_5"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="xboole_1.html#T64" onmouseover="rs('xboole_1/T64')" onmouseout="rh()">XBOOLE_1:64</a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">p</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T4" target="_self" onmouseover="rs('group_17/T4')" onmouseout="rh()">Th4</a>, <a class="txt" href="#E2:8_5"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E9:8"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E13:8"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="int_7.html#T15" onmouseover="rs('int_7/T15')" onmouseout="rh()">INT_7:15</a>, <a class="txt" href="#E10:8"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E11:8"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E12:8"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E2:8"><span class="lab"><font color="Green" title="E10">A2</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c11">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8_5"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</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">thus </span><font color="Maroon" title="c3">s</font> = 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c3">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
<br/>.= 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c1">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:8"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E16:8"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</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="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">t</font> = 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c4">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
<br/>.= 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c2">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:8"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="txt" href="#E17:8"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c2">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</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>
