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<span class="kw">let </span><font color="Maroon" title="c1">Q</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="quantal1.html#L3" title="QUANTAL1:struct.3">Girard-QuantaleStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Q</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">Q</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">Q</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c1">Q</font> is  <a href="quantal1.html#V20" title="QUANTAL1:attr.20">cyclic</a>  &amp; <font color="Maroon" title="c1">Q</font> is  <a href="quantal1.html#V21" title="QUANTAL1:attr.21">dualized</a>  ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">c</font> = the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font>;<br/>
<span class="kw">assume </span><a NAME="E1:53"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">Q</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">Q</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">Q</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">Q</font> is  <a href="quantal1.html#V20" title="QUANTAL1:attr.20">cyclic</a>  &amp; <font color="Maroon" title="c1">Q</font> is  <a href="quantal1.html#V21" title="QUANTAL1:attr.21">dualized</a>  )</span><br/>

<a NAME="E2:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">Q</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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a>, <a class="ref" href="zfmisc_1.html#T1" title="ZFMISC_1:th.1">ZFMISC_1:1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:53"/>
<font color="Maroon" title="c1">Q</font> is  <a href="quantal1.html#V20" title="QUANTAL1:attr.20">cyclic</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">Q</font> is  <a href="quantal1.html#V21" title="QUANTAL1:attr.21">dualized</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quantal1.html#D18" target="_self" title="QUANTAL1:def.18">QUANTAL1:def 18</a>,<a class="ref" href="quantal1.html#D19" target="_self" title="QUANTAL1:def.19">QUANTAL1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font></span><br/>

<a NAME="E1:53_1"/>
<font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:53_1"/>
<font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">Q</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="quantal1.html#D17" target="_self" title="QUANTAL1:def.17">QUANTAL1:def 17</a>,<a class="ref" href="quantal1.html#D20" target="_self" title="QUANTAL1:def.20">QUANTAL1:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )</span><br/>

<a NAME="E4:53"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:53"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font></span>)</span> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="quantal1.html#K2" title="QUANTAL1:func.2">-l&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font></span>)</span> <a href="quantal1.html#K1" title="QUANTAL1:func.1">-r&gt;</a> the <a href="quantal1.html#U1" title="QUANTAL1:sel.1">absurd</a> of <font color="Maroon" title="c1">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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