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<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#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>   st <font color="Olive" title="b1">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Olive" title="b2">q</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> $1 &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Olive" title="b1">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">q</font>;<br/>
<div><span class="lab"><font color="Green" title="E25">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Maroon" title="c3">q</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c3">q</font> implies <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E2:26_1_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c3">q</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E3:26_1_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:26_1_1"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_1_1"/>
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E2:26_1_1_1_1_1"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="int_7.html#T14" target="_self" title="INT_7:th.14">Th14</a></span>;<br/><a NAME="E3:26_1_1_1_1_1"/><span class="kw">then </span>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="int_7.html#T14" target="_self" title="INT_7:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:26_1_1_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="int_7.html#T4" target="_self" title="INT_7:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="newton.html#D6" title="NEWTON:def.6">NEWTON:def 6</a></span>;<br/><a NAME="E5:26_1_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="int_7.html#E10"><span class="lab"><font color="Green" title="E10">Lm4</font></span></a></span>;<br/><a NAME="E6:26_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="int_7.html#E14"><span class="lab"><font color="Green" title="E14">Lm7</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_1_1_2_1"/>
( <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> or <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">r1</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c6">p1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_1_1_2_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:26_1_1_1_1_2_1_1"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E6:26_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E35">A12</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E7:26_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:26_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E9:26_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="int_7.html#E23"><span class="lab"><font color="Green" title="E21">Lm10</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> or <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E2:26_1_1_1_1_2_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><a NAME="E3:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A17</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T120" title="XREAL_1:th.120">XREAL_1:120</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">q1</font>, <font color="Maroon" title="c9">r2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> <span class="kw"> such that </span><br/><a NAME="E6:26_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c8">q1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E7:26_1_1_1_1_2_1_1_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E8:26_1_1_1_1_2_1_1_1_1"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E9:26_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">and </span><br/><a NAME="E10:26_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
<font color="Maroon" title="c8">q1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:26_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A21</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E12:26_1_1_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E45">A22</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="int_7.html#E14"><span class="lab"><font color="Green" title="E14">Lm7</font></span></a>, <a class="txt" href="int_7.html#E23"><span class="lab"><font color="Green" title="E21">Lm10</font></span></a></span>;<br/><a NAME="E13:26_1_1_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E14:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><a NAME="E16:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E17:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A23</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E18:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A24</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E12:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><div><span class="lab"><font color="Green" title="E48">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1_2_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:26_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A26</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_1_1_2_1_1_1_1_1_1"/>
(  not <font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_1_1_1_1_1_1"/>
not <font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_1_1_1_2_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A27</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A26</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="txt" href="#E10:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a>, <a class="txt" href="#E18:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E47">A24</font></span></a>, <a class="txt" href="#E2:26_1_1_1_1_2_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A27</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A28</font></span>: 
<font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E37">A14</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">q1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E12:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="txt" href="#E17:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A21</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E49">A29</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">p</font> = 
 <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
;<br/><a NAME="E21:26_1_1_1_1_2_1_1_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="txt" href="#E12:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E45">A22</font></span></a>, <a class="txt" href="#E17:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E46">A23</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E22:26_1_1_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E49">A29</font></span></a>, <a class="txt" href="#E19:26_1_1_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E48">A25</font></span></a>, <a class="ref" href="funct_1.html#T166" title="FUNCT_1:th.166">FUNCT_1:166</a></span>;<br/><span class="kw">hence </span><a NAME="E23:26_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T3" target="_self" title="INT_7:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E39">A30</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E2:26_1_1_1_1_2_1_1_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><a NAME="E3:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A31</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T120" title="XREAL_1:th.120">XREAL_1:120</a></span>;<br/><a NAME="E5:26_1_1_1_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E6:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E40">A31</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><a NAME="E8:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E9:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A32</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E10:26_1_1_1_1_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">r2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b1">q1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Olive" title="b1">q1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">q1</font></span>)</span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">q1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E39">A30</font></span></a>, <a class="txt" href="int_7.html#E14"><span class="lab"><font color="Green" title="E14">Lm7</font></span></a>, <a class="txt" href="int_7.html#E22"><span class="lab"><font color="Green" title="E20">Lm9</font></span></a></span>;<br/><a NAME="E11:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E41">A32</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E12:26_1_1_1_1_2_1_1_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c5">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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E13:26_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E34">A33</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">r1</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> <span class="kw"> such that </span><br/><a NAME="E3:26_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E35">A34</font></span>: 
<font color="Maroon" title="c6">p1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E4:26_1_1_1_1_2_1_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:26_1_1_1_1_2_1_2"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c7">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E6:26_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E36">A35</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E7:26_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E37">A36</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:26_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E38">A37</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="int_7.html#E22"><span class="lab"><font color="Green" title="E20">Lm9</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_1_1_2_1_2_1"/>
( <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font> or <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_2_1_1"/><span class="lab"><font color="Green" title="E39">A38</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E2:26_1_1_1_1_2_1_2_1_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><a NAME="E3:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A39</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T120" title="XREAL_1:th.120">XREAL_1:120</a></span>;<br/><a NAME="E5:26_1_1_1_1_2_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E8:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E6:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><a NAME="E8:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E9:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A40</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E10:26_1_1_1_1_2_1_2_1_1"/>
 ex <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">r2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>  st <br/>( <font color="Olive" title="b1">q1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Olive" title="b1">q1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">q1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Olive" title="b1">q1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">q1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b1">q1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="txt" href="int_7.html#E14"><span class="lab"><font color="Green" title="E14">Lm7</font></span></a>, <a class="txt" href="int_7.html#E23"><span class="lab"><font color="Green" title="E21">Lm10</font></span></a></span>;<br/><a NAME="E11:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E8:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_2_1_1"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E12:26_1_1_1_1_2_1_2_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c5">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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E13:26_1_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font></span><br/><div class="add"><a NAME="E2:26_1_1_1_1_2_1_2_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><a NAME="E3:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A42</font></span>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T120" title="XREAL_1:th.120">XREAL_1:120</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">q1</font>, <font color="Maroon" title="c9">r2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> <span class="kw"> such that </span><br/><a NAME="E6:26_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c8">q1</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E7:26_1_1_1_1_2_1_2_1_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E8:26_1_1_1_1_2_1_2_1_2"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c9">r2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">and </span><br/><a NAME="E9:26_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E42">A44</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c8">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">and </span><br/><a NAME="E10:26_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E43">A45</font></span>: 
<font color="Maroon" title="c8">q1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c8">q1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:26_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E44">A46</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c8">q1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="txt" href="int_7.html#E14"><span class="lab"><font color="Green" title="E14">Lm7</font></span></a>, <a class="txt" href="int_7.html#E22"><span class="lab"><font color="Green" title="E20">Lm9</font></span></a></span>;<br/><a NAME="E12:26_1_1_1_1_2_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E8:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="ref" href="xreal_1.html#T74" title="XREAL_1:th.74">XREAL_1:74</a></span>;<br/><a NAME="E13:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><a NAME="E15:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T13" title="NEWTON:th.13">NEWTON:13</a></span>;<br/><a NAME="E16:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A47</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c6">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T10" title="NEWTON:th.10">NEWTON:10</a></span>;<br/><a NAME="E17:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E8:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E9:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E42">A44</font></span></a>, <a class="txt" href="#E11:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E18:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/><a NAME="E19:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A48</font></span>: 
<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>;<br/><a NAME="E20:26_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E47">A49</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c8">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E4:26_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E3:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E35">A34</font></span></a>, <a class="txt" href="#E8:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E38">A37</font></span></a>, <a class="txt" href="#E6:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E11:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E44">A46</font></span></a>, <a class="txt" href="#E16:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E45">A47</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><div><span class="lab"><font color="Green" title="E48">A50</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1_2_1_2_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:26_1_1_1_1_2_1_2_1_2_1"/><span class="lab"><font color="Green" title="E49">A51</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_1_1_1_1_2_1_2_1_2_1_1"/>
(  not <font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_2_1_2_1_1_1"/>
not <font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:26_1_1_1_1_2_1_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A52</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E36">A35</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_2_1_2_1"><span class="lab"><font color="Green" title="E49">A51</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">p1</font></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E37">A36</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E43">A45</font></span></a>, <a class="txt" href="#E20:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E47">A49</font></span></a>, <a class="txt" href="#E2:26_1_1_1_1_2_1_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E50">A52</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_2_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:26_1_1_1_1_2_1_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E50">A53</font></span>: 
<font color="Maroon" title="c10">z</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"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font> = 
<font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E34">A33</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="txt" href="#E1:26_1_1_1_1_2_1_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E22:26_1_1_1_1_2_1_2_1_2"/><span class="lab"><font color="Green" title="E49">A54</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E23:26_1_1_1_1_2_1_2_1_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1_1_1_2"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E24:26_1_1_1_1_2_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A55</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E46">A48</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">p</font> = 
 <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c2">p</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span> = 
<font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E48">A50</font></span></a>, <a class="ref" href="funct_1.html#T166" title="FUNCT_1:th.166">FUNCT_1:166</a></span>
<br/>.= 
<font color="Maroon" title="c3">q</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c3">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E49">A54</font></span></a>, <a class="txt" href="#E24:26_1_1_1_1_2_1_2_1_2"><span class="lab"><font color="Green" title="E50">A55</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/><span class="kw">hence </span><a NAME="E27:26_1_1_1_1_2_1_2_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T3" target="_self" title="INT_7:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:26_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:26_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:26"/><span class="lab"><font color="Green" title="E26">A56</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">q</font> implies <font color="Maroon" title="c1">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">q</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26_2"/><span class="lab"><font color="Green" title="E27">A57</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E2:26_2"/><span class="lab"><font color="Green" title="E28">A58</font></span>: 
<font color="Maroon" title="c2">q</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">and </span><br/><a NAME="E3:26_2"/><span class="lab"><font color="Green" title="E29">A59</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E4:26_2"/><span class="lab"><font color="Green" title="E30">A60</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c2">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">q</font></span><br/>

<a NAME="E5:26_2"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_2"><span class="lab"><font color="Green" title="E27">A57</font></span></a>, <a class="ref" href="int_7.html#T13" target="_self" title="INT_7:th.13">Th13</a></span>;<br/>
<a NAME="E6:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A61</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E7:26_2"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_2"><span class="lab"><font color="Green" title="E29">A59</font></span></a>, <a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></span>;<br/>
<a NAME="E8:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A62</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_2"><span class="lab"><font color="Green" title="E31">A61</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E9:26_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A63</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_2"><span class="lab"><font color="Green" title="E27">A57</font></span></a>, <a class="ref" href="int_7.html#T14" target="_self" title="INT_7:th.14">Th14</a></span>;<br/>
<a NAME="E10:26_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_2"><span class="lab"><font color="Green" title="E28">A58</font></span></a>, <a class="txt" href="#E4:26_2"><span class="lab"><font color="Green" title="E30">A60</font></span></a>, <a class="txt" href="#E8:26_2"><span class="lab"><font color="Green" title="E32">A62</font></span></a>, <a class="ref" href="int_7.html#T14" target="_self" title="INT_7:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:26_2"/>
<font color="Maroon" title="c1">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26_2"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="ref" href="int_7.html#T4" target="_self" title="INT_7:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:26"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E26">A56</font></span></a>, <a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E25">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:26"/>
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#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>   st <font color="Olive" title="b2">p</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp; <font color="Olive" title="b3">q</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a>  &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Olive" title="b1">n</font> &amp;  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Olive" title="b3">q</font> holds <br/><font color="Olive" title="b2">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
