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<span class="kw">let </span><font color="Maroon" title="c1">O</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">Ordinal</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">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">perm</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">O</font><br/>  for <font color="Olive" title="b3">s</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">O</font>,<font color="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b4">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">O</font> holds <br/> ( <font color="Olive" title="b4">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Olive" title="b2">perm</font></span>)</span> iff <font color="Olive" title="b4">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">perm</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">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">perm</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">O</font><br/>  for <font color="Olive" title="b2">s</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b3">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">O</font> holds <br/> ( <font color="Olive" title="b3">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Olive" title="b1">perm</font></span>)</span> iff <font color="Olive" title="b3">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">perm</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">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">perm</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <font color="Maroon" title="c1">O</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">s</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b2">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">O</font> holds <br/> ( <font color="Olive" title="b2">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span> iff <font color="Olive" title="b2">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">Series</a> of <font color="Maroon" title="c1">O</font>,<font color="Maroon" title="c2">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">  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">O</font> holds <br/> ( <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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span> iff <font color="Olive" title="b1">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">O</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="c5">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span> iff <font color="Maroon" title="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</font> )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c4">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font>;<br/>
<a NAME="E1:42"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></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> <font color="Maroon" title="c1">O</font> &amp;  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">s</font> )
 
<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/>
<a NAME="E2:42"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilb10_2.html#D4" target="_self" onmouseover="rs('hilb10_2/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:42"/>
( <font color="Maroon" title="c5">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span> implies <font color="Maroon" title="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</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="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</font> implies <font color="Maroon" title="c5">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1"/>
<font color="Maroon" title="c5">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></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="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</font></span><br/>

<a NAME="E2:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c5">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>
 
<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="E3:42_1"/>
<font color="Maroon" title="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">O</font>
 
<span class="kw">by</span> <span class="lab"><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="E4:42_1"/>
<font color="Maroon" title="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <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="E4:42"/>
<font color="Maroon" title="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</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">s</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="c5">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span></span><br/>

<a NAME="E5:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">perm</font></span>)</span> <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#D3" onmouseover="rs('polynom1/D3')" onmouseout="rh()">POLYNOM1:def 3</a></span>;<br/>
<a NAME="E6:42"/>
<font color="Maroon" title="c5">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">O</font>
 
<span class="kw">by</span> <span class="lab"><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="E7:42"/>
<font color="Maroon" title="c5">b</font> <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">s</font> <a href="hilb10_2.html#K4" title="HILB10_2:func.4">permuted_by</a> <font color="Maroon" title="c3">perm</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <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>
