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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  iff  ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:20"/>
( <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  implies  ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) implies <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_1"/>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  )</span><br/>

<a NAME="E2:20_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V3" title="WAYBEL_8:attr.3">arithmetic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="waybel15.html#E12"><span class="lab"><font color="Green" title="E10">Lm2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_1"/>
 ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E2:20"/>
(  ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  ) implies <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>  )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="20_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:20_2"/>
 ex <font color="Olive" title="b1">A</font> being   <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>   <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> ex <font color="Olive" title="b2">g</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">g</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a>  &amp; <font color="Olive" title="b2">g</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> </span><br/>

<a NAME="E2:20_2"/><span class="kw">then </span>
 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#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">p</font> being   <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</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#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Olive" title="b1">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">BoolePoset</a> <font color="Olive" title="b1">X</font></span>)</span> st <br/>( <font color="Olive" title="b2">p</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">p</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="waybel15.html#E16"><span class="lab"><font color="Green" title="E13">Lm3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:20_2"/>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="waybel15.html#E19"><span class="lab"><font color="Green" title="E16">Lm4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
