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<b>let </b><font color="Maroon" title="c1">n</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Olive" title="b3">r</font> implies <font color="Olive" title="b1">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Olive" title="b3">r</font> ) &amp; ( ( ( <font color="Olive" title="b1">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Olive" title="b3">r</font> ) or ( <font color="Olive" title="b1">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Olive" title="b3">r</font> ) or ( <font color="Olive" title="b1">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Olive" title="b3">r</font> ) ) implies <font color="Olive" title="b1">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Olive" title="b3">r</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> ) &amp; ( ( ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) ) implies <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) )</span><br/>



<b>thus </b><a NAME="E1:15"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) ) implies <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:15_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font>
 <b>and </b><br/><a NAME="E2:15_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">i</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> <b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E5:15_1"/><i><font color="Green" title="E12">A5</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>and </b><br/><a NAME="E6:15_1"/><i><font color="Green" title="E13">A6</font></i>: 
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="c5">i</font> holds <br/><font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">Def1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">j</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> <b> such that </b><br/><a NAME="E8:15_1"/><i><font color="Green" title="E14">A7</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <b>and </b><br/><a NAME="E9:15_1"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <b>and </b><br/><a NAME="E10:15_1"/><i><font color="Green" title="E16">A9</font></i>: 
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="c6">j</font> holds <br/><font color="Maroon" title="c3">q</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="#E2:15_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">t</font> =  <a href="xxreal_0.html#K3" title="XXREAL_0:func.3">min</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</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>  <b>by </b><i><a class="ref" href="finseq_2.html#T1" title="FINSEQ_2:th.1">FINSEQ_2:1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">t</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">POLYNOM3:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font> &amp; ( 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/><font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> ) )</span><br/>

<b>thus </b><a NAME="E12:15_1"/>
<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> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E8:15_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T15" title="XXREAL_0:th.15">XXREAL_0:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font> &amp; ( 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/><font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_1_1"/>
( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1_1"/><i><font color="Green" title="E17">A10</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><a NAME="E2:15_1_1_1_1"/><b>then </b><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:15_1_1_1_1"/>
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="#E4:15_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_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="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <b>by </b><i><a class="txt" href="#E5:15_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E10:15_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="#E1:15_1_1_1_1"><i><font color="Green" title="E17">A10</font></i></a>, <a class="txt" href="#E2:15_1_1_1_1"><i><font color="Green" title="E18">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_2"><b>suppose </b></a><a NAME="E1:15_1_1_1_2"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><a NAME="E2:15_1_1_1_2"/><b>then </b><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#D9" title="XXREAL_0:def.9">XXREAL_0:def 9</a></i>;<br/><a NAME="E3:15_1_1_1_2"/>
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="#E8:15_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1_2"/>
<font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <b>by </b><i><a class="txt" href="#E6:15_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="#E9:15_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="#E1:15_1_1_1_2"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="#E2:15_1_1_1_2"><i><font color="Green" title="E18">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1_3"><b>suppose </b></a><a NAME="E1:15_1_1_1_3"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font></span><br/><div class="add"><b>hence </b><a NAME="E2:15_1_1_1_3"/>
<font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 <b>by </b><i><a class="txt" href="#E5:15_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="#E9:15_1"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:15_1"/>
<font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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/><font color="Maroon" title="c2">p</font> <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> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font></span><br/>

<b>let </b><font color="Maroon" title="c8">k</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">t</font> implies <font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E15:15_1"/><i><font color="Green" title="E17">A14</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <b>and </b><br/><a NAME="E16:15_1"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">t</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/>

<a NAME="E17:15_1"/>
( <font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17" title="XXREAL_0:th.17">XXREAL_0:17</a></i>;<br/>
<a NAME="E18:15_1"/><b>then </b><i><font color="Green" title="E19">A16</font></i>: 
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> )
 
<b>by </b><i><a class="txt" href="#E16:15_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font> = 
<font color="Maroon" title="c3">q</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="#E6:15_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="#E15:15_1"><i><font color="Green" title="E17">A14</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">r</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
<b>by </b><i><a class="txt" href="#E10:15_1"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="#E15:15_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="#E18:15_1"><i><font color="Green" title="E19">A16</font></i></a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:15"/><i><font color="Green" title="E9">A17</font></i>: 
( ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_2"/>
( ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> ) or ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> ) )
 </span><b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E9">A17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_1"><b>suppose </b></a><a NAME="E1:15_2_1"/>
( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font></span><br/><div class="add"><a NAME="E2:15_2_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; ( <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> or <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> ) )
 <b>by </b><i><a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:15_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_2"><b>suppose </b></a><a NAME="E1:15_2_2"/>
( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font></span><br/><div class="add"><a NAME="E2:15_2_2"/><b>then </b>
( ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> or <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> ) &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R1" title="POLYNOM3:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> )
 <b>by </b><i><a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E3:15_2_2"/>
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_2_3"><b>suppose </b></a><a NAME="E1:15_2_3"/>
( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font></span><br/><div class="add"><b>hence </b><a NAME="E2:15_2_3"/>
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
