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<b>let </b><font color="Maroon" title="c1">o1</font>, <font color="Maroon" title="c2">o2</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">b1</font>, <font color="Olive" title="b2">c1</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">o1</font><br/> for <font color="Olive" title="b3">b2</font>, <font color="Olive" title="b4">c2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">o2</font>  st <font color="Olive" title="b1">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b2">c1</font> &amp; <font color="Olive" title="b3">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b4">c2</font> holds <br/><font color="Olive" title="b1">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b3">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b2">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b4">c2</font></span><br/><b>let </b><font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c4">c1</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">o1</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">b2</font>, <font color="Olive" title="b2">c2</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">o2</font>  st <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Olive" title="b1">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Olive" title="b2">c2</font> holds <br/><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b1">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Olive" title="b2">c2</font></span><br/><b>let </b><font color="Maroon" title="c5">b2</font>, <font color="Maroon" title="c6">c2</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c6">c2</font> implies <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font> )</span><br/>







<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c3">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Maroon" title="c5">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c6">c2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><b>reconsider </b><font color="Maroon" title="c8">b1'</font> = <font color="Maroon" title="c3">b1</font>, <font color="Maroon" title="c9">c1'</font> = <font color="Maroon" title="c4">c1</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o1</font> ;<br/><a NAME="E3:15_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c8">b1'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">c1'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_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"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o1</font> or <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">o1</font> )
 </span><b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1_1"><b>suppose </b></a><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">k'</font> = <font color="Maroon" title="c7">k</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E3:15_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k'</font>
 <b>by </b><i><a class="txt" href="#E1:15_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="#E3:15_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="#E1:15_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</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_2"><b>suppose </b></a><a NAME="E1:15_1_1_2"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">o1</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">o2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c1">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c10">k'</font> = <font color="Maroon" title="c7">k</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><a NAME="E3:15_1_1_2"/><i><font color="Green" title="E15">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k'</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c1">o1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:15_1_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>;<br/><a NAME="E4:15_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">k'</font> <a href="ordinal3.html#K5" title="ORDINAL3:func.5">-^</a> <font color="Maroon" title="c1">o1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:15_1_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="polynom6.html#D1" target="_self" title="POLYNOM6:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E3:15_1_1_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</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="E3:15"/>
<font color="Maroon" title="c3">b1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c5">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c4">c1</font> <a href="polynom6.html#K1" title="POLYNOM6:func.1">+^</a> <font color="Maroon" title="c6">c2</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T50" title="POLYNOM1:th.50">POLYNOM1:50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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