<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">k1</font>, <font color="Maroon" title="c4">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="Red">::  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="c2">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="c3">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">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="c2">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> implies <font color="Maroon" title="c4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">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="c3">k1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c3">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="E2:26"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/>

<span class="kw">assume </span><a NAME="E3:26"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c4">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">k1</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"> contradiction</span><br/>

<a NAME="E4:26"/>
<font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E5:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="polynom3.html#T11" target="_self" title="POLYNOM3:th.11">Th11</a></span>;<br/>
<a NAME="E6:26"/><span class="kw">then </span>
<font color="Maroon" title="c3">k1</font> <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="c4">k2</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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">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="E9:26"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">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="c5">A</font>
 <span class="kw">and </span><br/><a NAME="E10:26"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
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="c5">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="c2">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="E11:26"/>
 <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#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></span>;<br/>
<a NAME="E12:26"/><span class="kw">then </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="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a>, <a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></span>;<br/>
<a NAME="E13:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E21">A6</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="E14:26"/>
<font color="Maroon" title="c3">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:26_1"/>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T107" title="RVSUM_1:th.107">RVSUM_1:107</a></span>;<br/>
<a NAME="E2:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A9</font></span>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c4">k2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c4">k2</font></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:26_1"/>
<font color="Maroon" title="c3">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:26_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:26_1"/><span class="kw">then </span>
not <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a></span>;<br/>
<a NAME="E6:26_1"/><span class="kw">then </span>
not <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c2">n</font>,2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:26_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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="E15:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
<font color="Maroon" title="c3">k1</font> <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="c2">n</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 href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T107" title="RVSUM_1:th.107">RVSUM_1:107</a></span>
<br/>.= 
<font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:26"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>
;<br/>
<a NAME="E17:26"/><span class="kw">then </span>
<span class="p1"><a href="polynom3.html#K2" title="POLYNOM3:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="polynom3.html#K2" title="POLYNOM3:func.2">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E23">A8</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E19:26"/>
<font color="Maroon" title="c6">k</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="c2">n</font>,2</span>)</span>
 <span class="kw">and </span><br/><a NAME="E20:26"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K6" title="POLYNOM3:func.6">Decomp</a> <font color="Maroon" title="c2">n</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></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="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">k</font> as    <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">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E22:26"/>
<font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E23:26"/><span class="kw">then </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="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="polynom3.html#T11" target="_self" title="POLYNOM3:th.11">Th11</a></span>;<br/>
<a NAME="E24:26"/><span class="kw">then </span>
<font color="Maroon" title="c1">i</font> <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="c7">k</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/>
<span class="kw">hence </span><a NAME="E25:26"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E20:26"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="polynom3.html#T11" target="_self" title="POLYNOM3:th.11">Th11</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/>


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