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

<b>let </b><font color="Maroon">G1</font> be   <a href="group_1.html#NM1">Group</a>, <font color="Maroon">G2</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G1</font>,<font color="Maroon">G2</font>;<br/>



<a NAME="E1:34"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G1</font></span>)</span> holds  <span class="p1">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G1</font></span>)</span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11">lattice</a> <font color="Maroon">G1</font></span>)</span>;<br/>

<a NAME="E1:34_1"/><i><font color="Green">E23</font></i>: 
 <a href="group_4.html#K11">lattice</a> <font color="Maroon">G1</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="group_3.html#K1">Subgroups</a> <font color="Maroon">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K9">SubJoin</a> <font color="Maroon">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_4.html#K10">SubMeet</a> <font color="Maroon">G1</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="group_4.html#D13">GROUP_4:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">A1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E5:34_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="latsubgr.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>thus </b><a NAME="E6:34_1"/>
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><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">A1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:34_1_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><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">B1</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:34_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B1</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:34_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:34_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>
<a NAME="E6:34_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E8:34_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font></span>)</span>
 ;<br/>
<a NAME="E9:34_1_1"/><i><font color="Green">E42</font></i>: 
<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">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="latsubgr.html#T4" target="_self">Th4</a>, <a class="ref" href="latsubgr.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E10:34_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</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="Maroon">A1</font> holds <br/><font color="Olive">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</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="Maroon">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</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="Maroon">A1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:34_1_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:34_1_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:34_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E6:34_1_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:34_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T41">GROUP_6:41</a></i>;<br/>
<b>hence </b><a NAME="E8:34_1_1_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</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="Maroon">A1</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:34_1_1"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</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="Maroon">A1</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</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="Maroon">A1</font> holds <br/><font color="Olive">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> <a href="hidden.html#R2">in</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="Maroon">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>, <font color="Maroon">g2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</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="Maroon">A1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</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="Maroon">A1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:34_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:34_1_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E6:34_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:34_1_1_2"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:34_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A1</font>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<a NAME="E9:34_1_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:34_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:34_1_1_2"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</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="Maroon">A1</font>
 <b>by </b><i>, , <a class="txt" href="latsubgr.html#E1:34_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">A3</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E13:34_1_1"/><i><font color="Green">E92</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A3</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="Maroon">A1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:34_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="latsubgr.html#E5:34_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>
<a NAME="E14:34_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>
<a NAME="E15:34_1_1"/><b>then </b><i><font color="Green">E93</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E17:34_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font></span>)</span>
 ;<br/>
<a NAME="E18:34_1_1"/><i><font color="Green">E95</font></i>: 
<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">B1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="latsubgr.html#T4" target="_self">Th4</a>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E19:34_1_1"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</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="Maroon">B1</font> holds <br/><font color="Olive">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</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="Maroon">B1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</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="Maroon">B1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:34_1_1_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:34_1_1_3"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:34_1_1_3"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E6:34_1_1_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:34_1_1_3"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T41">GROUP_6:41</a></i>;<br/>
<b>hence </b><a NAME="E8:34_1_1_3"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</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="Maroon">B1</font>
 <b>by </b><i>, <a class="txt" href="latsubgr.html#E2:34_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:34_1_1"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</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="Maroon">B1</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</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="Maroon">B1</font> holds <br/><font color="Olive">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> <a href="hidden.html#R2">in</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="Maroon">B1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>, <font color="Maroon">g2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_4"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</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="Maroon">B1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</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="Maroon">B1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:34_1_1_4"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:34_1_1_4"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E2:34_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E6:34_1_1_4"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:34_1_1_4"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:34_1_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E3:34_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<a NAME="E9:34_1_1_4"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:34_1_1_4"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:34_1_1_4"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</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="Maroon">B1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E2:34_1_1"><i><font color="Green">E35</font></i></a>, , , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">B3</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E22:34_1_1"/><i><font color="Green">E108</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B3</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="Maroon">B1</font>
 <b>by </b><i>, <a class="txt" href="latsubgr.html#E1:34_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>
<a NAME="E23:34_1_1"/><i><font color="Green">E110</font></i>: 
 <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><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">B1</font></span>)</span> <a href="group_2.html#R1">=</a> <font color="Maroon">B3</font>
 
<b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">C1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E25:34_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">C1</font> <a href="group_2.html#R1">=</a> <font color="Maroon">A1</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B1</font>
 ;<br/>
<a NAME="E26:34_1_1"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>
<a NAME="E27:34_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E29:34_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font></span>)</span>
 ;<br/>
<a NAME="E30:34_1_1"/><i><font color="Green">E116</font></i>: 
<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">C1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="latsubgr.html#T4" target="_self">Th4</a>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E31:34_1_1"/><i><font color="Green">E117</font></i>: 
for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</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="Maroon">C1</font> holds <br/><font color="Olive">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</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="Maroon">C1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_5"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</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="Maroon">C1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:34_1_1_5"/><i><font color="Green">E118</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:34_1_1_5"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E8:34_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:34_1_1_5"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E6:34_1_1_5"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E7:34_1_1_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T41">GROUP_6:41</a></i>;<br/>
<b>hence </b><a NAME="E8:34_1_1_5"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</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="Maroon">C1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E8:34_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="latsubgr.html#E9:34_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:34_1_1"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</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="Maroon">C1</font> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</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="Maroon">C1</font> holds <br/><font color="Olive">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> <a href="hidden.html#R2">in</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="Maroon">C1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>, <font color="Maroon">g2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_6"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</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="Maroon">C1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</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="Maroon">C1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E3:34_1_1_6"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:34_1_1_6"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E9:34_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E6:34_1_1_6"/><i><font color="Green">E123</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E8:34_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:34_1_1_6"/>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:34_1_1_6"/><b>then </b>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">C1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E10:34_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<a NAME="E9:34_1_1_6"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E10:34_1_1_6"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:34_1_1_6"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</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="Maroon">C1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E9:34_1_1"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">C3</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G2</font><b> such that </b><br/><a NAME="E34:34_1_1"/><i><font color="Green">E125</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</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="Maroon">C1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E4:34_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="latsubgr.html#E6:34_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>
<a NAME="E35:34_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> <a href="xboole_0.html#K2">\/</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G1</font> <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <span class="p1">(<span class="default"><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">A1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><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">B1</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G2</font> ;<br/>
<a NAME="E38:34_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7"><b>proof </b></a><div class="add">

<a NAME="E1:34_1_1_7"/><i><font color="Green">E127</font></i>: 
<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">A1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:34_1_1_7_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Maroon">A1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E4:34_1_1_7_1"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E10:34_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:34_1_1_7_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:34_1_1_7_1"/><b>then </b>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T38">GROUP_4:38</a></i>;<br/>
<a NAME="E7:34_1_1_7_1"/><b>then </b>
<font color="Maroon">y</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="group_4.html#K5">gr</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:34_1_1_7_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:34_1_1_7_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:34_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="latsubgr.html#E8:34_1_1"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_1_1_7"/>
<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">B1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:34_1_1_7_2"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Maroon">B1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E4:34_1_1_7_2"/><i><font color="Green">E131</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E10:34_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<a NAME="E5:34_1_1_7_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:34_1_1_7_2"/><b>then </b>
<font color="Maroon">y</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T38">GROUP_4:38</a></i>;<br/>
<a NAME="E7:34_1_1_7_2"/><b>then </b>
<font color="Maroon">y</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="group_4.html#K5">gr</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E8:34_1_1_7_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:34_1_1_7_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:34_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="latsubgr.html#E8:34_1_1"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:34_1_1_7"/><b>then </b>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E9:34_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E4:34_1_1_7"/><b>then </b>
 <a href="group_4.html#K5">gr</a> <font color="Maroon">B</font> is    <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">C3</font>
 
<b>by </b><i><a class="ref" href="group_4.html#D5">GROUP_4:def 5</a></i>;<br/>
<a NAME="E5:34_1_1_7"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E6:34_1_1_7"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font>
 <b>by </b><i>, , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E7:34_1_1_7"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E8:34_1_1_7"/><b>then </b><i><font color="Green">E132</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A3</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T50">FUNCT_2:50</a></i>;<br/>
<a NAME="E9:34_1_1_7"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E10:34_1_1_7"/><b>then </b><i><font color="Green">E133</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B3</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T50">FUNCT_2:50</a></i>;<br/>
<a NAME="E11:34_1_1_7"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:34_1_1_7_3"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A3</font>
 ;<br/>

<a NAME="E2:34_1_1_7_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A3</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E3:34_1_1_7_3"/><b>then </b><i><font color="Green">E138</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:34_1_1_7_3"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E5:34_1_1_7_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="latsubgr.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E6:34_1_1_7_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1_1_7_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:34_1_1_7"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:34_1_1_7_4"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B3</font>
 ;<br/>

<a NAME="E2:34_1_1_7_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E3:34_1_1_7_4"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:34_1_1_7_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G2</font>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E5:34_1_1_7_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="latsubgr.html#T14" target="_self">Th14</a>, </i>;<br/>
<a NAME="E6:34_1_1_7_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1_1_7_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:34_1_1_7"/><b>then </b><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A3</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B3</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">AA</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A3</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">B3</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G1</font> ;<br/>
<a NAME="E15:34_1_1_7"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">AA</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E10:34_1_1"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E16:34_1_1_7"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55">GROUP_2:55</a></i>;<br/>
<a NAME="E17:34_1_1_7"/><b>then </b>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E18:34_1_1_7"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#T40">GROUP_6:40</a></i>;<br/>
<a NAME="E19:34_1_1_7"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E20:34_1_1_7"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> holds <br/><font color="Olive">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_7_5"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 ;<br/>

<a NAME="E2:34_1_1_7_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E3:34_1_1_7_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:34_1_1_7_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K3">"</a>  <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60">GROUP_2:60</a></i>;<br/>
<a NAME="E5:34_1_1_7_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="group_6.html#T41">GROUP_6:41</a></i>;<br/>
<a NAME="E6:34_1_1_7_5"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1_1_7_5"/>
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:34_1_1_7"/>
for <font color="Olive">g1</font>, <font color="Olive">g2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>  st <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> &amp; <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> holds <br/><font color="Olive">g1</font> <a href="group_1.html#K1">*</a> <font color="Olive">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>, <font color="Maroon">g2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font>;<br/>

<b>assume </b><a NAME="E1:34_1_1_7_6"/>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> &amp; <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> )
 ;<br/>

<a NAME="E2:34_1_1_7_6"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>
<a NAME="E3:34_1_1_7_6"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g2</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font> )
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:34_1_1_7_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">g2</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T59">GROUP_2:59</a></i>;<br/>
<a NAME="E5:34_1_1_7_6"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font></span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
 
<b>by </b><i><a class="ref" href="group_6.html#D7">GROUP_6:def 7</a></i>;<br/>
<a NAME="E6:34_1_1_7_6"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font></span>)</span> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:34_1_1_7_6"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T46">FUNCT_2:46</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">H</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E23:34_1_1_7"/><i><font color="Green">E146</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 <b>by </b><i><a class="ref" href="latsubgr.html#D1" target="_self">Def1</a>, , <a class="ref" href="group_2.html#T61">GROUP_2:61</a></i>;<br/>
<a NAME="E24:34_1_1_7"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="latsubgr.html#T14" target="_self">Th14</a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E25:34_1_1_7"/><b>then </b>
 <a href="group_4.html#K5">gr</a> <font color="Maroon">A</font> is    <a href="group_6.html#M1">Subgroup</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_4.html#D5">GROUP_4:def 5</a></i>;<br/>
<a NAME="E26:34_1_1_7"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E27:34_1_1_7"/><b>then </b><i><font color="Green">E147</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E3:34_1_1"><i><font color="Green">E36</font></i></a>, , </i>;<br/>
<a NAME="E28:34_1_1_7"/><i><font color="Green">E148</font></i>: 
<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">C1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_7_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:34_1_1_7_7"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="Maroon">C1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G2</font> ;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font><b> such that </b><br/><a NAME="E4:34_1_1_7_7"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T116">FUNCT_2:116</a></i>;<br/>
<b>thus </b><a NAME="E5:34_1_1_7_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:34_1_1_7"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T145">FUNCT_1:145</a></i>;<br/>
<b>hence </b><a NAME="E30:34_1_1_7"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C3</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font></span>)</span>
 <b>by </b><i><a class="txt" href="latsubgr.html#E8:34_1_1"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b> <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B1</font></span>)</span></span>)</span> = 
<font color="Maroon">A3</font> <a href="group_4.html#K8">"\/"</a> <font color="Maroon">B3</font>
<b>by </b><i><a class="txt" href="latsubgr.html#E3:34_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="latsubgr.html#E8:34_1_1"><i><font color="Green">E41</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2">(<span class="default"><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">A1</font></span>)</span></span>)</span> <a href="group_4.html#K8">"\/"</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2">(<span class="default"><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">B1</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="latsubgr.html#E2:34_1_1"><i><font color="Green">E35</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="latsubgr.html#T3" target="_self">Th3</a>, , </i>
;<br/>
<b>hence </b><a NAME="E40:34_1_1"/>
<span class="p1">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="latsubgr.html#E1:34_1"><i><font color="Green">E23</font></i></a>, </i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34"/>
 <a href="latsubgr.html#K3" target="_self">FuncLatt</a> <font color="Maroon">f</font> is    <a href="vectsp_8.html#M4">sup-Semilattice-Homomorphism</a> of  <a href="group_4.html#K11">lattice</a> <font color="Maroon">G1</font>, <a href="group_4.html#K11">lattice</a> <font color="Maroon">G2</font>
 <b>by </b><i><a class="ref" href="vectsp_8.html#D9">VECTSP_8:def 9</a></i>;<br/>


</div>
