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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L9</font> being    <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">L9</font> is  <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a>  holds <br/><font color="Olive" title="b1">L9</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">L9</font> be    <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">L9</font> is  <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a>  implies <font color="Maroon" title="c2">L9</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:18"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c2">L9</font> is  <a href="msualg_7.html#V1" title="MSUALG_7:attr.1">/\-inheriting</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">L9</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> </span><br/>

<a NAME="E2:18"/>
 for <font color="Olive" title="b1">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L9</font>  ex <font color="Olive" title="b2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Olive" title="b1">X</font> &amp; (  for <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>  st <font color="Olive" title="b3">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" 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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> &amp; (  for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>  st <font color="Olive" title="b2">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">a</font> =  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>);<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">a9</font> =  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>) as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="msualg_7.html#D2" target="_self" title="MSUALG_7:def.2">Def2</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">a9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a9</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> &amp; (  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">a9</font> ) )</span><br/>

<a NAME="E2:18_1"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>) <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_1"/>
<font color="Maroon" title="c5">a9</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#T12" target="_self" title="MSUALG_7:th.12">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">a9</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">b9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">b9</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c6">b9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">a9</font> )</span><br/>

<a NAME="E4:18_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_lat.html#D12" title="NAT_LAT:def.12">NAT_LAT:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:18_1"/>
<font color="Maroon" title="c6">b9</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">a9</font></span><br/>

<a NAME="E7:18_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#T12" target="_self" title="MSUALG_7:th.12">Th13</a></span>;<br/>
<a NAME="E8:18_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T39" title="LATTICE3:th.39">LATTICE3:39</a></span>;<br/>
<font color="Maroon" title="c6">b9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">a9</font> = 
<font color="Maroon" title="c7">b</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">L</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_7.html#T11" target="_self" title="MSUALG_7:th.11">Th12</a></span>
<br/>.= 
<font color="Maroon" title="c6">b9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:18_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:18_1"/>
<font color="Maroon" title="c6">b9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" title="LATTICES:th.4">LATTICES:4</a></span>; <a class="txt" 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">hence </span><a NAME="E3:18"/>
<font color="Maroon" title="c2">L9</font> is  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_8.html#D6" title="VECTSP_8:def.6">VECTSP_8:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
