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<b>reconsider </b><font color="Maroon" title="c3">O</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <span class="p1">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">L</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c4">O'</font> = <font color="Maroon" title="c3">O</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c5">O'</font> = <font color="Maroon" title="c4">O'</font> as   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="177_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c5">O'</font> implies not <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font> ) &amp; ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c5">O'</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c5">O'</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:177_1_1_1_1"/><i><font color="Green" title="E100">A1</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c5">O'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span><br/><b>then reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c6">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><b>assume </b><a NAME="E3:177_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <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"> contradiction</span><br/><b>then </b><font color="Maroon" title="c5">O'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c7">x'</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E5:177_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom1.html#E1:177_1_1_1_1"><i><font color="Green" title="E100">A1</font></i></a>, <a class="ref" href="polynom1.html#D9" target="_self" title="POLYNOM1:def.9">Def9</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><b>assume </b><a NAME="E2:177_1_1_1"/><i><font color="Green" title="E100">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c5">O'</font></span><br/><a NAME="E3:177_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E4:177_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">O'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</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>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:177_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="polynom1.html#T27" target="_self" title="POLYNOM1:th.27">Th27</a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E5:177_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c5">O'</font>
 <b>by </b><i><a class="txt" href="polynom1.html#E2:177_1_1_1"><i><font color="Green" title="E100">A2</font></i></a>, <a class="ref" href="polynom1.html#D9" target="_self" title="POLYNOM1:def.9">Def9</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>
<a NAME="E5:177_1_1"/><b>then </b>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c5">O'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K15" title="POLYNOM1:func.15">EmptyBag</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:177_1_1"/>
 <a href="polynom1.html#K26" title="POLYNOM1:func.26">1_</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font> is <a href="polynom1.html#V2" title="POLYNOM1:attr.2">finite-Support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D10" target="_self" title="POLYNOM1:def.10">Def10</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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