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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">C</font></span>)</span>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K6" title="HEYTING2:func.6">SUB</a> <font color="Olive" title="b3">u</font> holds <br/><font color="Olive" title="b2">v</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Olive" title="b3">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K6" title="HEYTING2:func.6">SUB</a> <font color="Olive" title="b2">u</font> holds <br/><font color="Olive" title="b1">v</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Olive" title="b2">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>; <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">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K6" title="HEYTING2:func.6">SUB</a> <font color="Maroon" title="c4">u</font> implies <font color="Maroon" title="c3">v</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:30"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting2.html#K6" title="HEYTING2:func.6">SUB</a> <font color="Maroon" title="c4">u</font>
 ; <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">v</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">u9</font> = <font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c6">v9</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c7">dv</font> = <span class="p1">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="substlat.html#D4" title="SUBSTLAT:def.4">SUBSTLAT:def 4</a></span>;<br/>
<a NAME="E3:30"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c4">u</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="heyting2.html#D7" target="_self" title="HEYTING2:def.7">Def7</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">v</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span> = 
the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <span class="p1">(<span class="default"><a href="substlat.html#K5" title="SUBSTLAT:func.5">SubstLatt</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">v</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting2.html#K7" title="HEYTING2:func.7">diff</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">v</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D1" title="LATTICES:def.1">LATTICES:def 1</a></span>
<br/>.= 
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v9</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <font color="Maroon" title="c7">dv</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="substlat.html#D4" title="SUBSTLAT:def.4">SUBSTLAT:def 4</a></span>
<br/>.= 
 <a href="substlat.html#K3" title="SUBSTLAT:func.3">mi</a> <font color="Maroon" title="c5">u9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></span>
<br/>.= 
<font color="Maroon" title="c4">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="substlat.html#T11" title="SUBSTLAT:th.11">SUBSTLAT:11</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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