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<span class="kw">consider </span><font color="Maroon" title="c4">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font><span class="kw"> such that </span><br/><a NAME="E2:20_1_1"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
for <font color="Olive" title="b1">b9</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom7.html#D4" target="_self" title="POLYNOM7:def.4">Def4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1_1_1"/>
( <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:20_1_1_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or (  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">X</font> ) )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:20_1_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or (  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">X</font> ) )
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:20_1_1_1_1_2"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or (  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">X</font> ) )</span><br/><div class="add"><a NAME="E2:20_1_1_1_1_2"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1_1_1_1_2_1"/>
 <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">sm</font> =  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">c</font> being    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of <font color="Maroon" title="c5">sm</font>;<br/>
<a NAME="E4:20_1_1_1_1_2_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font>
 
;<br/>
<a NAME="E5:20_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:20_1_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:20_1_1"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E1:20_1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A2</font></span></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><span class="kw">hence </span><a NAME="E3:20_1_1_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or (  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">X</font> ) )
 ; <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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:20_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c3">m</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font> or (  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <font color="Maroon" title="c1">X</font> ) )
 ; <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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