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<span class="kw">let </span><font color="Maroon" title="c1">U0</font> be   <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a>  <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</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">L</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="vectsp_8.html#D6" title="VECTSP_8:def.6">VECTSP_8:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1"/>
( <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
<font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:34_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:34_1_1_1"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font>
  <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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">l1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> or  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c3">l1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E1:34_1_1_1_1"/>
(  not <font color="Maroon" title="c3">l1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> or  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c3">l1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1"><span class="lab"><font color="Green" title="E18">A1</font></span></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/>


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

<span class="kw">let </span><font color="Maroon" title="c3">l2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">l2</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Maroon" title="c3">l2</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:34_1_1_1"/>
<font color="Maroon" title="c3">l2</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</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">l2</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E3:34_1_1_1"/>
<font color="Maroon" title="c3">l2</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T45" title="LATTICES:th.45">LATTICES:45</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/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_2"/>
<font color="Maroon" title="c2">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">H</font> = <font color="Maroon" title="c2">L</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font></span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">l1</font> =  <a href="unialg_3.html#K7" title="UNIALG_3:func.7">meet</a> <font color="Maroon" title="c3">H</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c4">l1</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="c4">l1</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">l1</font> ) ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">H</font>;<br/><span class="kw">thus </span><a NAME="E5:34_1_2"/>
<font color="Maroon" title="c4">l1</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font>
  <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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">l1</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">l2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> or <font color="Maroon" title="c4">l1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">l2</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">U1</font> = <font color="Maroon" title="c6">l2</font> as   <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c6">l2</font> as    <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font> ;<br/>
<span class="kw">assume </span><a NAME="E3:34_1_2_1"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c6">l2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</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="c4">l1</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">l2</font></span><br/>

<a NAME="E4:34_1_2_1"/>
<span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_3.html#D4" target="_self" title="UNIALG_3:def.4">Def4</a></span>;<br/>
<a NAME="E5:34_1_2_1"/><span class="kw">then </span>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font></span>)</span> <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="c7">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1_2_1"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a>, <a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></span>;<br/>
<a NAME="E6:34_1_2_1"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_3.html#K7" title="UNIALG_3:func.7">meet</a> <font color="Maroon" title="c3">H</font></span>)</span> <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="c7">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_3.html#D5" target="_self" title="UNIALG_3:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:34_1_2_1"/>
<font color="Maroon" title="c4">l1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_3.html#T17" target="_self" title="UNIALG_3:th.17">Th17</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/>


</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c6">l3</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">l3</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</font> or <font color="Maroon" title="c6">l3</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">l1</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">U1</font> = <font color="Maroon" title="c6">l3</font> as   <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></span>;<br/><span class="kw">assume </span><a NAME="E7:34_1_2"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c6">l3</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c2">L</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="c6">l3</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">l1</font></span><br/><a NAME="E8:34_1_2"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> holds <br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">A</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"> ( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:34_1_2_2"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">H1</font> = <font color="Maroon" title="c8">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">l4</font> being    <a href="unialg_3.html#M2" title="UNIALG_3:mode.2">Element</a> of  <a href="unialg_3.html#K1" title="UNIALG_3:func.1">Sub</a> <font color="Maroon" title="c1">U0</font><span class="kw"> such that </span><br/><a NAME="E4:34_1_2_2"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
( <font color="Maroon" title="c10">l4</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c9">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">l4</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_2_2"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">l4</font> = <font color="Maroon" title="c10">l4</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K12" title="UNIALG_2:func.12">UnSubAlLattice</a> <font color="Maroon" title="c1">U0</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">U2</font> = <font color="Maroon" title="c11">l4</font> as   <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_2.html#D15" title="UNIALG_2:def.15">UNIALG_2:def 15</a></span>;<br/>
<a NAME="E7:34_1_2_2"/>
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">U2</font> &amp; <font color="Maroon" title="c6">l3</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">l4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34_1_2"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E4:34_1_2_2"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="unialg_3.html#D4" target="_self" title="UNIALG_3:def.4">Def4</a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:34_1_2_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_3.html#T17" target="_self" title="UNIALG_3:th.17">Th17</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/>


</div><span class="kw">end;</span></div><a NAME="E9:34_1_2"/>
<span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></span>;<br/><a NAME="E10:34_1_2"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_3.html#K6" title="UNIALG_3:func.6">Carr</a> <font color="Maroon" title="c1">U0</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c3">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:34_1_2"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></span>;<br/><a NAME="E11:34_1_2"/><span class="kw">then </span>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">U1</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 <span class="p1">(<span class="default"><a href="unialg_3.html#K7" title="UNIALG_3:func.7">meet</a> <font color="Maroon" title="c3">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_3.html#D5" target="_self" title="UNIALG_3:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E12:34_1_2"/>
<font color="Maroon" title="c6">l3</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="unialg_3.html#T17" target="_self" title="UNIALG_3:th.17">Th17</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
