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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">k1</font>, <font color="Maroon" title="c5">k2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> implies ( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> iff <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:24"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E2:24"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> iff <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font> )</span><br/>

<a NAME="E3:24"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="funct_1.html#D2" title="FUNCT_1:def.2">FUNCT_1:def 2</a></span>;<br/>
<a NAME="E4:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c2">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">A</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:24"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
 <a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2</span>)</span>,<font color="Maroon" title="c6">A</font>)
 <span class="kw">and </span><br/><a NAME="E7:24"/>
 for <font color="Olive" title="b1">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> iff  <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D4" target="_self" title="POLYNOM3:def.4">Def4</a></span>;<br/>
<a NAME="E8:24"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T15" title="ORDERS_1:th.15">ORDERS_1:15</a></span>;<br/>
<a NAME="E9:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A6</font></span>: 
 <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2 <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c6">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T37" title="ORDERS_1:th.37">ORDERS_1:37</a>, <a class="ref" href="orders_1.html#T38" title="ORDERS_1:th.38">ORDERS_1:38</a></span>;<br/>
<a NAME="E10:24"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="funct_1.html#D2" title="FUNCT_1:def.2">FUNCT_1:def 2</a></span>;<br/>
<a NAME="E11:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A8</font></span>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> (<font color="Maroon" title="c3">n</font>,2)</span>)</span> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c1">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">thus </span><a NAME="E12:24"/>
( <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font> implies <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font></span><br/>

<a NAME="E2:24_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>,<span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/>
<a NAME="E3:24_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D3" target="_self" title="POLYNOM3:def.3">Def3</a></span>;<br/>
<a NAME="E4:24_1"/>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E1:24_1"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/>
<a NAME="E5:24_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:24_1"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">and </span><br/><a NAME="E8:24_1"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">and </span><br/><a NAME="E9:24_1"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">t</font> holds <br/><span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">Def1</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_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="E1:24_1_1"/>
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font></span><br/><div class="add"><a NAME="E2:24_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E1:24_1_1_1"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font></span><br/><div class="add"><a NAME="E2:24_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1"><span class="lab"><font color="Green" title="E27">A13</font></span></a></span>;<br/><a NAME="E3:24_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E4:24_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E5:24_1_1_2"/>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:24_1"><span class="lab"><font color="Green" title="E26">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E13:24"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font></span><br/>

<a NAME="E14:24"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
;<br/>
<a NAME="E15:24"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E16:24"/>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 &amp; <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<a NAME="E17:24"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E14:24"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">Def1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E18:24"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2"><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="E1:24_2"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1"><span class="kw">suppose </span></a><a NAME="E1:24_2_1"/>
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:24_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E13:24"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E15:24"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_2"><span class="kw">suppose </span></a><a NAME="E1:24_2_2"/><span class="lab"><font color="Green" title="E28">A20</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:24_2_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>,<span class="p2"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/><a NAME="E3:24_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A21</font></span>: 
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D3" target="_self" title="POLYNOM3:def.3">Def3</a></span>;<br/><a NAME="E4:24_2_2"/>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c5">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k1</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E9:24"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E10:24"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E3:24"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E11:24"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E4:24"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E1:24_2_2"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:24_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:24"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="txt" href="#E3:24_2_2"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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