<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">VS</font> being <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font> holds   <a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Olive" title="b1">VS</font> is <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a></span><br/><b>let </b><font color="Maroon" title="c2">VS</font> be  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a> <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of <font color="Maroon" title="c1">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font> is <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a></span><br/><b>let </b><font color="Maroon" title="c3">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_8.html#D6" target="_self" title="VECTSP_8:def.6">VECTSP_8:def 6</a> <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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> 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">Y</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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">a</font> ) )</span><br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( <font color="Maroon" title="c3">Y</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="c3">Y</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="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">Y</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:()"><i><font color="Red">::  thesis: </font></i></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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> 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">Y</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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">a</font> ) )</span><br/><div class="add"><b>thus </b><a NAME="E2:14_1_1"/>
 ex <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> 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">Y</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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">a</font> ) )
  <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="14_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b1">b</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="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> ) )</span><br/>

<b>thus </b><a NAME="E1:14_1_1_1"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</font>
  <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">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b1">b</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="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font> )</span><br/>

<b>thus </b><a NAME="E1:14_1_1_1_1"/>
(  not <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c4">q</font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:14_1_1"><i><font color="Green" title="E9">A1</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>

<b>let </b><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></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="c4">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c4">b</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="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E2:14_1_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">b</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="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span></span><br/>

<b>thus </b><a NAME="E3:14_1_1_1"/>
<font color="Maroon" title="c4">b</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="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#T45" title="LATTICES:th.45">LATTICES:45</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></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/>
<font color="Maroon" title="c3">Y</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:()"><i><font color="Red">::  thesis: </font></i></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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> 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">Y</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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b2">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive" title="b1">a</font> ) )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">X</font> = <font color="Maroon" title="c3">Y</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="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c5">p</font> =  <a href="vectsp_8.html#K5" title="VECTSP_8:func.5">meet</a> <font color="Maroon" title="c4">X</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/><b>take </b>
<font color="Maroon" title="c5">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</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 <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font> ) )</span><br/><b>thus </b><a NAME="E4:14_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</font>
  <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">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>  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">Y</font> holds <br/><font color="Olive" title="b1">b</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c7">H</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E2:14_1_2_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c6">q</font> as    <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font> ;<br/>
<b>assume </b><a NAME="E4:14_1_2_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">q</font></span><br/>

<a NAME="E5:14_1_2_1"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">h</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">H</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E2:14_1_2_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self" title="VECTSP_8:def.4">Def4</a></i>;<br/>
<a NAME="E6:14_1_2_1"/><b>then </b>
 <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="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</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">H</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:14_1_2_1"><i><font color="Green" title="E10">A3</font></i></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></i>;<br/>
<a NAME="E7:14_1_2_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K5" title="VECTSP_8:func.5">meet</a> <font color="Maroon" title="c4">X</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">H</font>
 
<b>by </b><i><a class="ref" href="vectsp_8.html#D5" target="_self" title="VECTSP_8:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E8:14_1_2_1"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E2:14_1_2_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="vectsp_8.html#T6" target="_self" title="VECTSP_8:th.6">Th6</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>let </b><font color="Maroon" title="c6">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></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="c6">r</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</font> implies <font color="Maroon" title="c6">r</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font> )</span><br/><b>assume </b><a NAME="E5:14_1_2"/><i><font color="Green" title="E9">A4</font></i>: 
<font color="Maroon" title="c6">r</font> <a href="lattice3.html#R3" title="LATTICE3:pred.3">is_less_than</a> <font color="Maroon" title="c3">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">r</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font></span><br/><b>consider </b><font color="Maroon" title="c7">H</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E7:14_1_2"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><a NAME="E9:14_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
for <font color="Olive" title="b1">Z1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font> holds <br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">Z1</font> be    <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"> ( <font color="Maroon" title="c9">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font> implies the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font> )</span><br/>

<b>assume </b><a NAME="E1:14_1_2_2"/><i><font color="Green" title="E12">A7</font></i>: 
<font color="Maroon" title="c9">Z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>
 ; <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="c7">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c10">Z2</font> = <font color="Maroon" title="c9">Z1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">VS</font> ;<br/>
<b>consider </b><font color="Maroon" title="c11">z1</font> being    <a href="vectsp_8.html#M2" title="VECTSP_8:mode.2">Element</a> of  <a href="vectsp_8.html#K2" title="VECTSP_8:func.2">Subspaces</a> <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E4:14_1_2_2"/><i><font color="Green" title="E13">A8</font></i>: 
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c10">Z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">z1</font> )
 <b>by </b><i><a class="txt" href="vectsp_8.html#E1:14_1_2_2"><i><font color="Green" title="E12">A7</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">z1</font> = <font color="Maroon" title="c11">z1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</span> ;<br/>
<b>consider </b><font color="Maroon" title="c13">z3</font> being  <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>  <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c2">VS</font><b> such that </b><br/><a NAME="E7:14_1_2_2"/><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c13">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z1</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3" title="VECTSP_5:def.3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E8:14_1_2_2"/><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c9">Z1</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="c13">z3</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E4:14_1_2_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="vectsp_8.html#E7:14_1_2_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="ref" href="vectsp_8.html#D4" target="_self" title="VECTSP_8:def.4">Def4</a></i>;<br/>
<a NAME="E9:14_1_2_2"/>
<font color="Maroon" title="c6">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">z1</font>
 
<b>by </b><i><a class="txt" href="vectsp_8.html#E5:14_1_2"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="vectsp_8.html#E4:14_1_2_2"><i><font color="Green" title="E13">A8</font></i></a>, <a class="ref" href="lattice3.html#D16" title="LATTICE3:def.16">LATTICE3:def 16</a></i>;<br/>
<b>hence </b><a NAME="E10:14_1_2_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E7:14_1_2"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="vectsp_8.html#E7:14_1_2_2"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="vectsp_8.html#E8:14_1_2_2"><i><font color="Green" title="E15">A10</font></i></a>, <a class="ref" href="vectsp_8.html#T6" target="_self" title="VECTSP_8:th.6">Th6</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><a NAME="E10:14_1_2"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><a NAME="E11:14_1_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H</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="vectsp_8.html#K4" title="VECTSP_8:func.4">carr</a> <font color="Maroon" title="c2">VS</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E9:14_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="setfam_1.html#T6" title="SETFAM_1:th.6">SETFAM_1:6</a></i>;<br/><a NAME="E12:14_1_2"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">H</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="vectsp_8.html#K5" title="VECTSP_8:func.5">meet</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D5" target="_self" title="VECTSP_8:def.5">Def5</a></i>;<br/><b>hence </b><a NAME="E13:14_1_2"/>
<font color="Maroon" title="c6">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="vectsp_8.html#E7:14_1_2"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="vectsp_8.html#T6" target="_self" title="VECTSP_8:th.6">Th6</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></div><b>end;</b></div>

</div>
