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<b>let </b><font color="Maroon" title="c1">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"> for <font color="Olive" title="b1">L</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/> for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">L</font> &amp; ( for <font color="Olive" title="b3">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Olive" title="b2">x</font> holds <br/><span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b3">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="Olive" title="b1">L</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive" title="b2">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Olive" title="b1">x</font> holds <br/><span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b2">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/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</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"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x</font> holds <br/><span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">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/>





 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">L</font></span>)</span> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D24" title="POLYNOM1:def.24">POLYNOM1:def 24</a></i>
<br/>.= 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c1">X</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
;<br/>
<b>hence </b><a NAME="E2:15"/>
<span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">L</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>; <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">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x</font> holds <br/><span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Olive" title="b1">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/>

<b>let </b><font color="Maroon" title="c4">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x</font> implies <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</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/>

<b>assume </b><a NAME="E3:15"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x</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"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</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/>

<b>hence </b><span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c4">b</font> = 
<span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">L</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c4">b</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="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>
;<br/> <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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