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<b>let </b><font color="Maroon" title="c1">W</font> be  <a href="setfam_1.html#NV3" title="SETFAM_1:NV.3">with_non-empty_element</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">L</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> holds <br/>( <font color="Olive" title="b1">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span> iff ( <font color="Olive" title="b1">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Olive" title="b1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Olive" title="b1">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">L</font> be   <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"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font> implies ( <font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span> iff ( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> ) ) )</span><br/>



<b>assume </b><a NAME="E1:64"/><i><font color="Green" title="E26">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">W</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span> iff ( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> implies <font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span> )</span>
<div class="add"><b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )</span><br/><b>then reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">L</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span> ;<br/><a NAME="E3:64_1"/>
( <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K7" title="YELLOW21:func.7">-CONT_category</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="altcat_2.html#T30" title="ALTCAT_2:th.30">ALTCAT_2:30</a></i>;<br/><b>hence </b><a NAME="E4:64_1"/>
( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> )
 <b>by </b><i><a class="txt" href="#E1:64"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="yellow21.html#D12" target="_self" title="YELLOW21:def.12">Def12</a>, <a class="ref" href="yellow21.html#T17" target="_self" title="YELLOW21:th.17">Th17</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>

<b>assume </b><a NAME="E3:64"/><i><font color="Green" title="E27">A2</font></i>: 
( <font color="Maroon" title="c2">L</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp; <font color="Maroon" title="c2">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"> <font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span></span><br/>

<a NAME="E4:64"/><b>then </b>
<font color="Maroon" title="c2">L</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</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>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">L</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K7" title="YELLOW21:func.7">-CONT_category</a> </span>)</span> <b>by </b><i><a class="txt" href="#E1:64"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="yellow21.html#T17" target="_self" title="YELLOW21:th.17">Th17</a></i>;<br/>
<a NAME="E6:64"/>
 <a href="yellow21.html#K4" title="YELLOW21:func.4">latt</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">L</font>
 
;<br/>
<b>hence </b><a NAME="E7:64"/>
<font color="Maroon" title="c2">L</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">W</font> <a href="yellow21.html#K8" title="YELLOW21:func.8">-ALG_category</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E3:64"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="yellow21.html#D12" target="_self" title="YELLOW21:def.12">Def12</a></i>; <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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