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<b>let </b><font color="Maroon">F</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="vectsp_1.html#V11">strict</a> <a href="vectsp_1.html#NM6">VectSp</a> of <font color="Maroon">F</font>;<br/>



<b>set </b><font color="Maroon">f</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>;<br/>
<a NAME="E1:23"/><i><font color="Green">E24</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:23"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">A</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="vectsp_8.html#K2" target="_self">Subspaces</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">X1</font> being  <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:23_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<a NAME="E5:23_1"/><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:23_1"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:23_1_1"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> ;<br/>
<b>consider </b><font color="Maroon">Z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:23_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>thus </b><a NAME="E5:23_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:23_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 ;<br/>

<a NAME="E2:23_1_1_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E4:23_1_1_2"/>
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>
<b>hence </b><a NAME="E5:23_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X1</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:23_1"/>
<span class="p1">(<span class="default"><a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="vectsp_7.html#T16">VECTSP_7:16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:23"/>
 <a href="vectsp_8.html#K6" target="_self">FuncLatt</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp_8.html#K1" target="_self">lattice</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>;<br/>


</div>
