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

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</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">  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#NM9" title="VECTSP_1:NM.9">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/><span class="kw">let </span><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#NM9" title="VECTSP_1:NM.9">VectSp</a> of <font color="Maroon" title="c1">F</font>; <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">  <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/><span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="vectsp_8.html#D6" target="_self" onmouseover="rs('vectsp_8/D6')" onmouseout="rh()">VECTSP_8:def 6</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="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"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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"><span class="kw">suppose </span></a><a NAME="E1:14_1_1"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<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" 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="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"><span class="kw">thus </span><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" 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="14_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="vectsp_8.html#K1" title="VECTSP_8:func.1">lattice</a> <font color="Maroon" title="c2">VS</font></span>)</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"> (  <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/>

<span class="kw">thus </span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_1"><span class="lab"><font color="Green" title="E9">A1</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">  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/>

<span class="kw">let </span><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" 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="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/>

<span class="kw">assume </span><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" 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="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/>

<span class="kw">thus </span><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T19" onmouseover="rs('lattices/T19')" onmouseout="rh()">LATTICES:19</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><span class="kw">suppose </span></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" 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="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"><span class="kw">then </span><span class="kw">reconsider </span><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/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c5">p</font>
; <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="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/><span class="kw">set </span><font color="Maroon" title="c6">x</font> =  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">X</font>;<br/><span class="kw">thus </span><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" 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">  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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lattice3.html#D16" onmouseover="rs('lattice3/D16')" onmouseout="rh()">LATTICE3:def 16</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"> (  not <font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI: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="c7">q</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E2:14_1_2_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c7">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/>
<span class="kw">assume </span><a NAME="E4:14_1_2_1"/><span class="lab"><font color="Green" title="E10">A3</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 ; <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="c5">p</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c7">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="c9">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="c8">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_2_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="vectsp_8.html#D4" target="_self" onmouseover="rs('vectsp_8/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E6:14_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="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#K7" title="RELSET_1:func.7">.:</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="c8">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:14_1_2_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a>, <a class="ref" href="setfam_1.html#T3" onmouseover="rs('setfam_1/T3')" onmouseout="rh()">SETFAM_1:3</a></span>;<br/>
<a NAME="E7:14_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="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="c8">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_8.html#D5" target="_self" onmouseover="rs('vectsp_8/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><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="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1_2_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="vectsp_8.html#T6" target="_self" onmouseover="rs('vectsp_8/T6')" onmouseout="rh()">Th6</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><span class="kw">let </span><font color="Maroon" title="c7">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" 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="c7">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="c7">r</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c8">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><span class="kw"> such that </span><br/><a NAME="E6:14_1_2"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/><span class="kw">assume </span><a NAME="E7:14_1_2"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c7">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" 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="c7">r</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c5">p</font></span><br/><a NAME="E8:14_1_2"/><span class="lab"><font color="Green" title="E11">A6</font></span>: 
 for <font color="Olive" title="b1">Z1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Z1</font> <a href="tarski.html#R2" title="TARSKI: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#K7" title="RELSET_1:func.7">.:</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="c8">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="14_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">Z1</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c9">Z1</font> <a href="tarski.html#R2" title="TARSKI: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#K7" title="RELSET_1:func.7">.:</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="c8">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:14_1_2_2"/><span class="lab"><font color="Green" title="E12">A7</font></span>: 
<font color="Maroon" title="c9">Z1</font> <a href="tarski.html#R2" title="TARSKI: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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>
 ; <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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:14_1_2_2"/><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c11">z1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E5:14_1_2_2"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:14_1_2_2"><span class="lab"><font color="Green" title="E12">A7</font></span></a>, <a class="ref" href="funct_2.html#T65" onmouseover="rs('funct_2/T65')" onmouseout="rh()">FUNCT_2:65</a></span>;<br/>
<span class="kw">reconsider </span><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/>
<a NAME="E7:14_1_2_2"/><span class="lab"><font color="Green" title="E15">A10</font></span>: 
<font color="Maroon" title="c7">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:14_1_2"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="txt" href="#E4:14_1_2_2"><span class="lab"><font color="Green" title="E13">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:14_1_2_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c13">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_5.html#D3" onmouseover="rs('vectsp_5/D3')" onmouseout="rh()">VECTSP_5:def 3</a></span>;<br/>
<a NAME="E10:14_1_2_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:14_1_2_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a>, <a class="txt" href="#E9:14_1_2_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="vectsp_8.html#D4" target="_self" onmouseover="rs('vectsp_8/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:14_1_2_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14_1_2"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="txt" href="#E9:14_1_2_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="txt" href="#E7:14_1_2_2"><span class="lab"><font color="Green" title="E15">A10</font></span></a>, <a class="ref" href="vectsp_8.html#T6" target="_self" onmouseover="rs('vectsp_8/T6')" onmouseout="rh()">Th6</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><a NAME="E9:14_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>  the    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">X</font> <a href="tarski.html#R2" title="TARSKI: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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T35" onmouseover="rs('funct_2/T35')" onmouseout="rh()">FUNCT_2:35</a></span>;<br/><a NAME="E10:14_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="c8">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#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c4">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:14_1_2"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="setfam_1.html#T5" onmouseover="rs('setfam_1/T5')" onmouseout="rh()">SETFAM_1:5</a></span>;<br/><a NAME="E11:14_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="c8">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_8.html#D5" target="_self" onmouseover="rs('vectsp_8/D5')" onmouseout="rh()">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E12:14_1_2"/>
<font color="Maroon" title="c7">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:14_1_2"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="vectsp_8.html#T6" target="_self" onmouseover="rs('vectsp_8/T6')" onmouseout="rh()">Th6</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><span class="kw">end;</span></div>

</div>
