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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> ;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green" title="E10">A1</font></i>: 
not <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>
 ;<br/>

<a NAME="E2:17"/><b>then </b><i><font color="Green" title="E11">A2</font></i>: 
( not <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="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> )
 
<b>by </b><i><a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">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="E4:17"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">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="E5:17"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>and </b><br/><a NAME="E6:17"/><i><font color="Green" title="E14">A5</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="c4">i</font> holds <br/><font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_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="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="polynom3.html#E1:17"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="polynom3.html#T6" target="_self" title="POLYNOM3:th.6">Th6</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">i</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><br/>

<b>thus </b><a NAME="E7:17"/>
<font color="Maroon" title="c4">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>by </b><i><a class="txt" href="polynom3.html#E4:17"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>

<a NAME="E8:17"/>
<font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 
<b>by </b><i><a class="txt" href="polynom3.html#E2:17"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="polynom3.html#E4:17"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="polynom3.html#E6:17"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="polynom3.html#D1" target="_self" title="POLYNOM3:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E9:17"/>
<font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="polynom3.html#E5:17"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>

<b>thus </b><a NAME="E10:17"/>
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="c4">i</font> holds <br/><font color="Maroon" title="c3">q</font> <a href="wsierp_1.html#K1" title="WSIERP_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="c2">p</font> <a href="wsierp_1.html#K1" title="WSIERP_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="polynom3.html#E6:17"><i><font color="Green" title="E14">A5</font></i></a></i>;<br/>


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