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<span class="kw">let </span><font color="Maroon" title="c2">n</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">f</font> being   <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>   <a href="pre_poly.html#V2" title="PRE_POLY:attr.2">finite-support</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Olive" title="b1">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  holds <br/> <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> <font color="Olive" title="b1">f</font></span>)</span> <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">let </span><font color="Maroon" title="c3">f</font> be   <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>   <a href="pre_poly.html#V2" title="PRE_POLY:attr.2">finite-support</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font> &amp; <font color="Maroon" title="c3">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a>  implies  <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> <font color="Maroon" title="c3">f</font></span>)</span> <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">assume </span><span class="kw">that </span><br/><a NAME="E1:71"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E2:71"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c3">f</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</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="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> <font color="Maroon" title="c3">f</font></span>)</span> <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="c4">b</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> <font color="Maroon" title="c3">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">C</font> =  <a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</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"><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> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E4:71"/><span class="lab"><font color="Green" title="E58">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> <font color="Maroon" title="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K20" title="RVSUM_1:func.20">Product</a> <font color="Maroon" title="c6">F</font>
 <span class="kw">and </span><br/><a NAME="E5:71"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<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> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D5" onmouseover="rs('nat_3/D5')" onmouseout="rh()">NAT_3:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as   <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E59">A4</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">S</font> =  <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c7">F</font>;<br/>
<a NAME="E7:71"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K21" title="RVSUM_1:func.21">Product</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T10" onmouseover="rs('euler_2/T10')" onmouseout="rh()">EULER_2:10</a>, <a class="ref" href="rfinseq2.html#D6" onmouseover="rs('rfinseq2/D6')" onmouseout="rh()">RFINSEQ2:def 6</a></span>;<br/>
<a NAME="E8:71"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="number08.html#R1" title="NUMBER08:pred.1">&lt;=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="number08.html#T40" target="_self" onmouseover="rs('number08/T40')" onmouseout="rh()">Th40</a></span>;<br/>
<a NAME="E9:71"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c7">F</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="number08.html#T41" target="_self" onmouseover="rs('number08/T41')" onmouseout="rh()">Th41</a></span>;<br/>
<a NAME="E10:71"/><span class="lab"><font color="Green" title="E63">A8</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c7">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq2.html#D6" onmouseover="rs('rfinseq2/D6')" onmouseout="rh()">RFINSEQ2:def 6</a>, <a class="ref" href="classes1.html#T75" onmouseover="rs('classes1/T75')" onmouseout="rh()">CLASSES1:75</a></span>;<br/>
<a NAME="E11:71"/>
 <a href="rfinseq2.html#K6" title="RFINSEQ2:func.6">sort_a</a> <font color="Maroon" title="c7">F</font> is  <a href="valued_0.html#V7" title="VALUED_0:attr.7">increasing</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E5:71"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="number08.html#T43" target="_self" onmouseover="rs('number08/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:71"/>
 <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> <font color="Maroon" title="c3">f</font></span>)</span> <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="#E4:71"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E7:71"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E8:71"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="txt" href="#E9:71"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E10:71"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="number08.html#T35" target="_self" onmouseover="rs('number08/T35')" onmouseout="rh()">Th35</a>, <a class="ref" href="number08.html#T38" target="_self" onmouseover="rs('number08/T38')" onmouseout="rh()">Th38</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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