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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  st  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font></span><br/><b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</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"> (  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x2</font> implies <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E12">A1</font></i>: 
 <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x2</font>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

1 = 
<span class="p1">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c3">x2</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">x1</font>
<b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="hilbasis.html#T9" target="_self" title="HILBASIS:th.9">Th9</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E2:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="hilbasis.html#T9" target="_self" title="HILBASIS:th.9">Th9</a></i>
;<br/>
<b>hence </b><a NAME="E4:12"/>
contradiction
 ; <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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