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<b>let </b><font color="Maroon">F1</font> be   <a href="funct_2.html#NM1">Function</a> of 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>,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">B</font></span>)</span>, <font color="Maroon">F2</font> be   <a href="funct_2.html#NM1">Function</a> of 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>,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">B</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:16_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">S</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">A</font><br/> for <font color="Olive">B0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font>  st <font color="Olive">B0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">S</font> holds <br/><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Olive">B0</font>
 <b>and </b><br/><a NAME="E2:16_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">S</font> being <a href="vectsp_1.html#V11">strict</a>  <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">A</font><br/> for <font color="Olive">B0</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font>  st <font color="Olive">B0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">S</font> holds <br/><font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">S</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Olive">B0</font>
 ;<br/>

<a NAME="E3:16_1_2"/>
for <font color="Olive">h</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">h</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/><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_2_1"/>
<font color="Maroon">h</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="E2:16_1_2_1"/><b>then </b>
<font color="Maroon">h</font> is    <a href="vectsp_8.html#M2" target="_self">Element</a> of  <a href="vectsp_8.html#K2" target="_self">Subspaces</a> <font color="Maroon">A</font>
 
;<br/>
<b>then consider </b><font color="Maroon">S</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="E4:16_1_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>by </b><i><a class="ref" href="vectsp_5.html#D3">VECTSP_5:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B0</font> = <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">B</font> ;<br/>
<a NAME="E6:16_1_2_1"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="vectsp_7.html#K1">Lin</a> <font color="Maroon">B0</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:16_1_2_1"/>
<font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_1_2"/>
<font color="Maroon">F1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


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