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<span class="kw">let </span><font color="Maroon" title="c1">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">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b4">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b4">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b2">m</font></span>)</span></span>)</span> where <font color="Olive" title="b4">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">m</font>,<font color="Olive" title="b4">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">X</font> holds <br/><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b3">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b2">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> where <font color="Olive" title="b3">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Olive" title="b1">m</font>,<font color="Olive" title="b3">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b2">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b1">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">m</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">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">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b1">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">XX</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,  <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>] <span class="kw">means </span>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,  <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>] <span class="kw">means </span>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<span class="kw">set </span><font color="Maroon" title="c6">X1</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">X2</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> ;<br/>
<div><span class="lab"><font color="Green" title="E62">A1</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b3">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b3">p9</font>] </span> } </span>  &amp; <font color="Olive" title="b2">k2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b3">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b3">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">k1</font>, <font color="Maroon" title="c9">k2</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="c8">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c9">k2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_1"/>
( <font color="Maroon" title="c8">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c9">k2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k2</font></span><br/><a NAME="E2:68_1_1"/><span class="kw">then </span>
(  ex <font color="Olive" title="b1">p1</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k1</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p1</font>] ) &amp;  ex <font color="Olive" title="b1">p2</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p2</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p2</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p2</font>] ) )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_1"/>
<font color="Maroon" title="c8">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</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="lab"><font color="Green" title="E63">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_2"><span class="kw">now </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">  not <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:68_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> contradiction</span><br/><a NAME="E2:68_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_2"/>
contradiction
 ; <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">set </span><font color="Maroon" title="c8">m1</font> = <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,  <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>] <span class="kw">means </span>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ,  <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>] <span class="kw">means </span>( <font color="Maroon">P<sub>1</sub></font>[$1,$2,$3] &amp; $3 <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default">$3 <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> $2</span>)</span> );<br/>
<span class="kw">set </span><font color="Maroon" title="c9">X11</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">X12</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> ;<br/>
<div><span class="lab"><font color="Green" title="E64">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_3"><span class="kw">now </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">  for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b3">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b3">p9</font>] </span> } </span>  &amp; <font color="Olive" title="b2">k2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b3">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b3">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">k1</font>, <font color="Maroon" title="c12">k2</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">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c12">k2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_3"/>
( <font color="Maroon" title="c11">k1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  &amp; <font color="Maroon" title="c12">k2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k2</font></span><br/><a NAME="E2:68_1_3"/><span class="kw">then </span>
(  ex <font color="Olive" title="b1">p1</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p1</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p1</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k1</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p1</font>] ) &amp;  ex <font color="Olive" title="b1">p2</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p2</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p2</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k2</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p2</font>] ) )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_3"/>
<font color="Maroon" title="c11">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k2</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"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_4"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:68_1_4"/><span class="lab"><font color="Green" title="E65">A4</font></span>: 
( <font color="Maroon" title="c12">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">p9</font>] )
 ;<br/><a NAME="E4:68_1_4"/>
( ( <font color="Maroon" title="c12">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">p9</font>] ) or ( <font color="Maroon" title="c12">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">p9</font>] ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_4"><span class="lab"><font color="Green" title="E65">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:68_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  )
 ;<br/><span class="kw">hence </span><a NAME="E6:68_1_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0: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>
<a NAME="E5:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A5</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
;<br/>
<span class="kw">assume </span><a NAME="E6:68_1"/><span class="lab"><font color="Green" title="E66">A6</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> (  not <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font> )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_5"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">y9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:68_1_5"/><span class="lab"><font color="Green" title="E67">B1</font></span>: 
( <font color="Maroon" title="c12">y9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">y9</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E66">A6</font></span></a></span>;<br/><a NAME="E4:68_1_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">y9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon" title="c12">y9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">y9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T16" target="_self" onmouseover="rs('nat_4/T16')" onmouseout="rh()">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E5:68_1_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_5"><span class="lab"><font color="Green" title="E67">B1</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</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>
<a NAME="E8:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A7</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_6"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_6"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:68_1_6"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_6"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI: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>
<a NAME="E10:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A8</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
;<br/>
<a NAME="E11:68_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b4">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p9</font>] </span> } </span>  holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_4.html#S1" target="_self" onmouseover="rs('nat_4/S1')" onmouseout="rh()">NAT_4:sch 1</a>();<br/></span>
<a NAME="E12:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A9</font></span>: 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E66">A6</font></span></a></span>;<br/>
<a NAME="E13:68_1"/><span class="lab"><font color="Green" title="E70">A10</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A11</font></span>: <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c1">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T32" onmouseover="rs('nat_3/T32')" onmouseout="rh()">NAT_3:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:68_1"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="ref" href="nat_3.html#T22" onmouseover="rs('nat_3/T22')" onmouseout="rh()">NAT_3:22</a></span>
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_8"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_8"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E2:68_1_8"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_8"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E66">A6</font></span></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="E16:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A12</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
;<br/>
<a NAME="E17:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A13</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a></span>;<br/>
<a NAME="E18:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">a13</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  is  <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">included_in_Seg</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" onmouseover="rs('finseq_1/D13')" onmouseout="rh()">FINSEQ_1:def 13</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_9"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_9"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span><br/><a NAME="E2:68_1_9"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_9"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E20:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A14</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
;<br/>
<a NAME="E21:68_1"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:68_1"><span class="lab"><font color="Green" title="E73">A13</font></span></a></span>;<br/>
<a NAME="E22:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">a15</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  is  <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">included_in_Seg</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" onmouseover="rs('finseq_1/D13')" onmouseout="rh()">FINSEQ_1:def 13</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_10"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_10"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span><br/><a NAME="E2:68_1_10"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_10"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A16</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
;<br/>
<a NAME="E25:68_1"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:68_1"><span class="lab"><font color="Green" title="E73">A13</font></span></a></span>;<br/>
<a NAME="E26:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">a17</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  is  <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">included_in_Seg</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" onmouseover="rs('finseq_1/D13')" onmouseout="rh()">FINSEQ_1:def 13</a></span>;<br/>
<a NAME="E27:68_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:68_1"><span class="lab"><font color="Green" title="E77">A16</font></span></a>, <a class="txt" href="#E20:68_1"><span class="lab"><font color="Green" title="E75">A14</font></span></a>, <a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E79">A18</font></span>:  <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"> { <span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span> = 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>
<br/>.= 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:68_1"><span class="lab"><font color="Green" title="E78">a17</font></span></a>, <a class="txt" href="#E22:68_1"><span class="lab"><font color="Green" title="E76">a15</font></span></a>, <a class="txt" href="#E3:68_1"><span class="lab"><font color="Green" title="E64">A3</font></span></a>, <a class="ref" href="finseq_3.html#T42" onmouseover="rs('finseq_3/T42')" onmouseout="rh()">FINSEQ_3:42</a></span>
<br/>.= 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T51" target="_self" onmouseover="rs('nat_4/T51')" onmouseout="rh()">Th50</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" onmouseover="rs('rvsum_1/T75')" onmouseout="rh()">RVSUM_1:75</a></span>
;<br/>
<a NAME="E29:68_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b4">X2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p9</font>] </span> } </span>  &amp;  not <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="Olive" title="b3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">X2</font> holds <br/><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_4.html#S2" target="_self" onmouseover="rs('nat_4/S2')" onmouseout="rh()">NAT_4:sch 2</a>();<br/></span>
<a NAME="E30:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A19</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"> { <span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E63">A2</font></span></a></span>;<br/>
<a NAME="E31:68_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b4">X2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p9</font>] </span> } </span>  holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X2</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_4.html#S1" target="_self" onmouseover="rs('nat_4/S1')" onmouseout="rh()">NAT_4:sch 1</a>();<br/></span>
<a NAME="E32:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A20</font></span>: 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"> { <span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_12"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_12"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:68_1_12"/><span class="lab"><font color="Green" title="E82">A21</font></span>: 
( <font color="Maroon" title="c12">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">p9</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E66">A6</font></span></a></span>;<br/><a NAME="E4:68_1_12"/>
( ( <font color="Maroon" title="c12">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">p9</font>] ) or ( <font color="Maroon" title="c12">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c12">p9</font>] ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68_1_12"><span class="lab"><font color="Green" title="E82">A21</font></span></a></span>;<br/><a NAME="E5:68_1_12"/><span class="kw">then </span>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  )
 ;<br/><span class="kw">hence </span><a NAME="E6:68_1_12"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0: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>
<a NAME="E34:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A22</font></span>: 
<font color="Maroon" title="c4">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_13"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )</span><br/><span class="kw">assume </span><a NAME="E1:68_1_13"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font></span><br/><a NAME="E2:68_1_13"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_13"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:68_1"><span class="lab"><font color="Green" title="E66">A6</font></span></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="E36:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A23</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
;<br/>
<a NAME="E37:68_1"/><span class="kw">then </span>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c3">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:68_1"><span class="lab"><font color="Green" title="E67">A7</font></span></a></span>;<br/>
<a NAME="E38:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">a24</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  is  <a href="finseq_1.html#V3" title="FINSEQ_1:attr.3">included_in_Seg</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D13" onmouseover="rs('finseq_1/D13')" onmouseout="rh()">FINSEQ_1:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E85">A25</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1_14"><span class="kw">now </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">  not <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:68_1_14"/>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</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"> contradiction</span><br/><a NAME="E2:68_1_14"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p9</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] )
 ;<br/><span class="kw">hence </span><a NAME="E3:68_1_14"/>
contradiction
 ; <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="E41:68_1"/>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</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="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span><br/>

<a NAME="E42:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:68_1"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="txt" href="#E2:68_1"><span class="lab"><font color="Green" title="E63">A2</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/>
<a NAME="E43:68_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:68_1"><span class="lab"><font color="Green" title="E65">A5</font></span></a>, <a class="txt" href="#E40:68_1"><span class="lab"><font color="Green" title="E85">A25</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/>
<a NAME="E44:68_1"/><span class="kw">then </span>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b2">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b2">p9</font>] </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E45:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A26</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> 
 
;<br/>
<a NAME="E46:68_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b4">X11</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p9</font>] </span> } </span>  &amp;  not <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="Olive" title="b3">m</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">X11</font> holds <br/><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X11</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_4.html#S2" target="_self" onmouseover="rs('nat_4/S2')" onmouseout="rh()">NAT_4:sch 2</a>();<br/></span>
<a NAME="E47:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E87">A27</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"> { <span class="default"> <span class="p4">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:68_1"><span class="lab"><font color="Green" title="E85">A25</font></span></a></span>;<br/>
<a NAME="E48:68_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><br/>  for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b3">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b4">X11</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b4">X11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b5">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Olive" title="b3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b5">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b2">n</font>,<font color="Olive" title="b3">m</font>,<font color="Olive" title="b5">p9</font>] </span> } </span>  holds <br/> <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Olive" title="b4">X11</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_4.html#S1" target="_self" onmouseover="rs('nat_4/S1')" onmouseout="rh()">NAT_4:sch 1</a>();<br/></span>
<a NAME="E49:68_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A28</font></span>: 
 <a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2"> { <span class="default"> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
;<br/>
<a NAME="E50:68_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:68_1"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="txt" href="#E36:68_1"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="ref" href="xboole_1.html#T13" onmouseover="rs('xboole_1/T13')" onmouseout="rh()">XBOOLE_1:13</a></span>;<br/>
<span class="kw">then </span> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3">(<span class="default"><span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:68_1"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>
<br/>.= 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:68_1"><span class="lab"><font color="Green" title="E74">a13</font></span></a>, <a class="txt" href="#E38:68_1"><span class="lab"><font color="Green" title="E84">a24</font></span></a>, <a class="txt" href="#E1:68_1"><span class="lab"><font color="Green" title="E62">A1</font></span></a>, <a class="ref" href="finseq_3.html#T42" onmouseover="rs('finseq_3/T42')" onmouseout="rh()">FINSEQ_3:42</a></span>
<br/>.= 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T51" target="_self" onmouseover="rs('nat_4/T51')" onmouseout="rh()">Th50</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T75" onmouseover="rs('rvsum_1/T75')" onmouseout="rh()">RVSUM_1:75</a></span>
;<br/>
<span class="kw">then </span><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:68_1"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="ref" href="nat_4.html#T53" target="_self" onmouseover="rs('nat_4/T53')" onmouseout="rh()">Th52</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:68_1"><span class="lab"><font color="Green" title="E81">A20</font></span></a>, <a class="txt" href="#E30:68_1"><span class="lab"><font color="Green" title="E80">A19</font></span></a>, <a class="ref" href="nat_4.html#T53" target="_self" onmouseover="rs('nat_4/T53')" onmouseout="rh()">Th52</a></span>
<br/>.= 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p4"> { <span class="default"> <span class="p5">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="newton.html#K11" title="NEWTON:func.11">|^</a> <span class="p0">(<span class="default"><font color="Olive" title="b1">p9</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span> where <font color="Olive" title="b1">p9</font> is   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c3">m</font>,<font color="Olive" title="b1">p9</font>] </span> } </span> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:68_1"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="txt" href="#E49:68_1"><span class="lab"><font color="Green" title="E88">A28</font></span></a>, <a class="txt" href="#E47:68_1"><span class="lab"><font color="Green" title="E87">A27</font></span></a>, <a class="ref" href="nat_4.html#T53" target="_self" onmouseover="rs('nat_4/T53')" onmouseout="rh()">Th52</a></span>
<br/>.= 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:68_1"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="txt" href="#E10:68_1"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>
<br/>.= 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_4.html#K1" title="NAT_4:func.1">|-count</a> <span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">m1</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T44" onmouseover="rs('finseq_3/T44')" onmouseout="rh()">FINSEQ_3:44</a></span>
<br/>.= 
 <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">Sum</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">m1</font></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_4.html#T52" target="_self" onmouseover="rs('nat_4/T52')" onmouseout="rh()">Th51</a></span>
<br/>.= 
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c11">m1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T73" onmouseover="rs('rvsum_1/T73')" onmouseout="rh()">RVSUM_1:73</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E53:68_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K2" title="WSIERP_1:func.2">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <font color="Maroon" title="c4">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:68_1"><span class="lab"><font color="Green" title="E71">A11</font></span></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>
