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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><br/>  for <font color="Olive" title="b2">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/>  for <font color="Olive" title="b3">p</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b2">L</font> holds <br/> ( <font color="Olive" title="b1">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> iff <font color="Olive" title="b1">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Olive" title="b3">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> <br/>  for <font color="Olive" title="b2">p</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds <br/> ( <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> iff <font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Olive" title="b2">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">L</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L2" title="STRUCT_0:struct.2">ZeroStr</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font> holds <br/> ( <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> iff <font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Olive" title="b1">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> iff <font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c4">p</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">B</font> = <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a> ;<br/>
<a NAME="E1:27"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" onmouseover="rs('hilbasis/D1')" onmouseout="rh()">HILBASIS:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E10">A1</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> = 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">(<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c1">n</font>) <a href="bagorder.html#K1" title="BAGORDER:func.1">-cut</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_2.html#D3" target="_self" onmouseover="rs('hilb10_2/D3')" onmouseout="rh()">Def3</a></span>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c2">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_2.html#T5" target="_self" onmouseover="rs('hilb10_2/T5')" onmouseout="rh()">Th5</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E3:27"/>
( <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> implies <font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c4">p</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="27_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:27_2"/>
<font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c4">p</font></span><br/>

<a NAME="E2:27_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c2">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="c3">L</font> &amp; <font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" onmouseover="rs('polynom1/D3')" onmouseout="rh()">POLYNOM1:def 3</a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="pre_poly.html#D12" onmouseover="rs('pre_poly/D12')" onmouseout="rh()">PRE_POLY:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:27_2"/>
<font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" onmouseover="rs('polynom1/D3')" onmouseout="rh()">POLYNOM1:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:27"/>
<font color="Maroon" title="c2">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <font color="Maroon" title="c4">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span></span><br/>

<a NAME="E5:27"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c2">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="c3">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" onmouseover="rs('polynom1/D3')" onmouseout="rh()">POLYNOM1:def 3</a></span>;<br/>
<a NAME="E6:27"/>
( <font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:27"/>
<font color="Maroon" title="c2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom1.html#K3" title="POLYNOM1:func.3">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="hilb10_2.html#K3" title="HILB10_2:func.3">extended_by_0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" onmouseover="rs('polynom1/D3')" onmouseout="rh()">POLYNOM1:def 3</a>, <a class="txt" href="#E5:27"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E10">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
