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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> implies  ex <font color="Olive" title="b1">X</font> being 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>  ex <font color="Olive" title="b2">S</font> being <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> st <br/>( <font color="Olive" title="b2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> &amp; <font color="Olive" title="b2">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> &amp; <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  ) ) &amp; (  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">S</font> being <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> st <br/>( <font color="Olive" title="b2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> &amp; <font color="Olive" title="b2">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> &amp; <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  ) implies <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> ) )</span><br/>

<b>thus </b><a NAME="E1:40"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> implies  ex <font color="Olive" title="b1">X</font> being 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>  ex <font color="Olive" title="b2">S</font> being <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> st <br/>( <font color="Olive" title="b2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> &amp; <font color="Olive" title="b2">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> &amp; <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  ) )
 <b>by </b><i><a class="txt" href="waybel13.html#E26"><i><font color="Green" title="E22">Lm2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">S</font> being <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> st <br/>( <font color="Olive" title="b2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> &amp; <font color="Olive" title="b2">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> &amp; <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  ) implies <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )</span><br/>

<b>thus </b><a NAME="E2:40"/>
(  ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">S</font> being <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> st <br/>( <font color="Olive" title="b2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> &amp; <font color="Olive" title="b2">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> &amp; <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  ) implies <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1"/>
 ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">S</font> being <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font> st <br/>( <font color="Olive" title="b2">S</font> is <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> &amp; <font color="Olive" title="b2">S</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> &amp; <font color="Maroon" title="c1">L</font>,<font color="Olive" title="b2">S</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a></span><br/>

<a NAME="E2:40_1"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">c</font> being <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <font color="Olive" title="b1">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">c</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp; <font color="Maroon" title="c1">L</font>, <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b2">c</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )
 
<b>by </b><i><a class="txt" href="waybel13.html#E29"><i><font color="Green" title="E25">Lm4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:40_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>
 <b>by </b><i><a class="txt" href="waybel13.html#E37"><i><font color="Green" title="E33">Lm6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


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