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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">p</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Olive" title="b3">r</font> holds <br/><font color="Olive" title="b1">p</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</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="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c2">p</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font> )</span><br/>



<b>assume </b><a NAME="E1:86"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c3">q</font> <a href="polynom1.html#R1" title="POLYNOM1: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="polynom1.html#R1" title="POLYNOM1:pred.1">&lt;</a> <font color="Maroon" title="c4">r</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">k</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:86"/><i><font color="Green" title="E48">A2</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">k</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">k</font>
 <b>and </b><br/><a NAME="E4:86"/><i><font color="Green" title="E49">A3</font></i>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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">l</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">l</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D11" target="_self" title="POLYNOM1:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">m</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:86"/><i><font color="Green" title="E50">A4</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">m</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">m</font>
 <b>and </b><br/><a NAME="E7:86"/><i><font color="Green" title="E51">A5</font></i>: 
for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</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">l</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">l</font>
 <b>by </b><i><a class="txt" href="#E1:86"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="polynom1.html#D11" target="_self" title="POLYNOM1:def.11">Def11</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c5">k</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c6">m</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="polynom1.html#D11" target="_self" title="POLYNOM1:def.11">POLYNOM1:def 11</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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</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">n</font> &amp; ( for <font color="Olive" title="b1">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">n</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">l</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">l</font> ) )</span><br/>

<a NAME="E8:86"/><i><font color="Green" title="E52">A6</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c7">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">m</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E9:86"/><i><font color="Green" title="E53">A7</font></i>: 
( ( <font color="Maroon" title="c7">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font> implies <font color="Maroon" title="c7">n</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">k</font> ) &amp; ( <font color="Maroon" title="c7">n</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c5">k</font> implies ( <font color="Maroon" title="c7">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">k</font> ) ) &amp; ( <font color="Maroon" title="c7">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">m</font> implies <font color="Maroon" title="c7">n</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c6">m</font> ) &amp; ( <font color="Maroon" title="c7">n</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c6">m</font> implies ( <font color="Maroon" title="c7">n</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">m</font> ) ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_1_1"/>
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</font> or <font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</font> or <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">ORDINAL1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_1_1_1"><b>suppose </b></a><a NAME="E1:86_1_1_1"/>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">m</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">n</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">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:86_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">n</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">n</font>
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="#E7:86"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="#E9:86"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a>, <a class="ref" href="ordinal3.html#T16" title="ORDINAL3:th.16">ORDINAL3:16</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</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="86_1_1_2"><b>suppose </b></a><a NAME="E1:86_1_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">k</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">n</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">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:86_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">n</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">n</font>
 <b>by </b><i><a class="txt" href="#E4:86"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="#E6:86"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="#E9:86"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a>, <a class="ref" href="ordinal3.html#T16" title="ORDINAL3:th.16">ORDINAL3:16</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</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="86_1_1_3"><b>suppose </b></a><a NAME="E1:86_1_1_3"/>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</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">n</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">n</font></span><br/><div class="add"><b>hence </b><a NAME="E2:86_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">n</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">n</font>
 <b>by </b><i><a class="txt" href="#E3:86"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="#E6:86"><i><font color="Green" title="E50">A4</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="E11:86"/>
<font color="Maroon" title="c2">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c7">n</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">n</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">l</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">n</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">l</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">l</font></span><br/>

<b>let </b><font color="Maroon" title="c8">l</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">n</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">l</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">l</font> )</span><br/>

<b>assume </b><a NAME="E12:86"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c8">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">n</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">l</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">l</font></span><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">l</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">l</font>
<b>by </b><i><a class="txt" href="#E4:86"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E52">A6</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">l</font>
<b>by </b><i><a class="txt" href="#E7:86"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="#E8:86"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="#E12:86"><i><font color="Green" title="E54">A8</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>
