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

<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>;<br/>



<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ;<br/>

<a NAME="E2:31"/>
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x2</font> )
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font></span>)</span> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K11" title="GROUP_4:func.11">lattice</a> <font color="Maroon" title="c1">G1</font></span>)</span> )
 
;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Maroon" title="c1">G1</font> )
 
<b>by </b><i><a class="ref" href="group_4.html#T92" title="GROUP_4:th.92">GROUP_4:92</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x1</font> as  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">x2</font> = <font color="Maroon" title="c5">x2</font> as  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="group_3.html#D1" title="GROUP_3:def.1">GROUP_3:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">A1</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c9">A2</font> = <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">G2</font> ;<br/>
<a NAME="E7:31_1"/><i><font color="Green" title="E19">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c8">A1</font> &amp; <span class="p1">(<span class="default"><a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Maroon" title="c9">A2</font> )
 
<b>by </b><i><a class="ref" href="latsubgr.html#D3" target="_self" title="LATSUBGR:def.3">Def3</a></i>;<br/>
<a NAME="E8:31_1"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:31_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55" title="GROUP_2:th.55">GROUP_2:55</a></i>;<br/>
<a NAME="E10:31_1"/><b>then </b><i><font color="Green" title="E21">A6</font></i>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font> ;<br/>
<a NAME="E12:31_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E8:31_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="latsubgr.html#E10:31_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E13:31_1"/><i><font color="Green" title="E23">A8</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>;<br/>

<b>assume </b><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E3:31_1_1"/><i><font color="Green" title="E24">A9</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:31_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E5:31_1_1"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/>
<a NAME="E6:31_1_1"/><b>then </b><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E7:31_1_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T41" title="GROUP_6:th.41">GROUP_6:41</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_1"/>
<font color="Maroon" title="c11">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="latsubgr.html#E6:31_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:31_1"/>
for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>;<br/>

<b>assume </b><a NAME="E1:31_1_2"/><i><font color="Green" title="E24">A11</font></i>: 
( <font color="Maroon" title="c11">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E3:31_1_2"/><i><font color="Green" title="E25">A12</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:31_1_2"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E6:31_1_2"/><i><font color="Green" title="E27">A14</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:31_1_2"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:31_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E6:31_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E8:31_1_2"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E4:31_1_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>
<a NAME="E9:31_1_2"/><b>then </b><i><font color="Green" title="E28">A15</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E10:31_1_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:31_1_2"/>
<font color="Maroon" title="c11">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_2"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="latsubgr.html#E6:31_1_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="latsubgr.html#E9:31_1_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c11">B1</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G2</font><b> such that </b><br/><a NAME="E16:31_1"/><i><font color="Green" title="E24">A16</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E12:31_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="latsubgr.html#E13:31_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="group_2.html#T61" title="GROUP_2:th.61">GROUP_2:61</a></i>;<br/>
<a NAME="E17:31_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T55" title="GROUP_2:th.55">GROUP_2:55</a></i>;<br/>
<a NAME="E18:31_1"/><b>then </b><i><font color="Green" title="E25">A17</font></i>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font><b> such that </b><br/><a NAME="E20:31_1"/><i><font color="Green" title="E26">A18</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span>
 ;<br/>
<a NAME="E21:31_1"/><i><font color="Green" title="E27">A19</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="latsubgr.html#E8:31_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="latsubgr.html#E18:31_1"><i><font color="Green" title="E25">A17</font></i></a>, <a class="txt" href="latsubgr.html#E20:31_1"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E22:31_1"/><i><font color="Green" title="E28">A20</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> holds <br/><font color="Olive" title="b1">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">g</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>;<br/>

<b>assume </b><a NAME="E1:31_1_3"/>
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E3:31_1_3"/><i><font color="Green" title="E29">A21</font></i>: 
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:31_1_3"/>
<font color="Maroon" title="c14">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_3"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E5:31_1_3"/><b>then </b>
<font color="Maroon" title="c14">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T60" title="GROUP_2:th.60">GROUP_2:60</a></i>;<br/>
<a NAME="E6:31_1_3"/><b>then </b><i><font color="Green" title="E30">A22</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E7:31_1_3"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">x</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> 
 
<b>by </b><i><a class="ref" href="group_6.html#T41" title="GROUP_6:th.41">GROUP_6:41</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_3"/>
<font color="Maroon" title="c13">g</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_3"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="latsubgr.html#E6:31_1_3"><i><font color="Green" title="E30">A22</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:31_1"/>
for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">g1</font>, <font color="Maroon" title="c14">g2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>;<br/>

<b>assume </b><a NAME="E1:31_1_4"/><i><font color="Green" title="E29">A23</font></i>: 
( <font color="Maroon" title="c13">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c15">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E3:31_1_4"/><i><font color="Green" title="E30">A24</font></i>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c13">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E4:31_1_4"/><i><font color="Green" title="E31">A25</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E6:31_1_4"/><i><font color="Green" title="E32">A26</font></i>: 
( <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">y</font> )
 <b>by </b><i><a class="txt" href="latsubgr.html#E1:31_1_4"><i><font color="Green" title="E29">A23</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E7:31_1_4"/>
<font color="Maroon" title="c16">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E6:31_1_4"><i><font color="Green" title="E32">A26</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E8:31_1_4"/><b>then </b>
<font color="Maroon" title="c15">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E4:31_1_4"><i><font color="Green" title="E31">A25</font></i></a>, <a class="ref" href="group_2.html#T59" title="GROUP_2:th.59">GROUP_2:59</a></i>;<br/>
<a NAME="E9:31_1_4"/><b>then </b><i><font color="Green" title="E33">A27</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E10:31_1_4"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a></i>;<br/>
<b>hence </b><a NAME="E11:31_1_4"/>
<font color="Maroon" title="c13">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E3:31_1_4"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="latsubgr.html#E6:31_1_4"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="latsubgr.html#E9:31_1_4"><i><font color="Green" title="E33">A27</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c13">B2</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G2</font><b> such that </b><br/><a NAME="E25:31_1"/><i><font color="Green" title="E29">A28</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="latsubgr.html#E21:31_1"><i><font color="Green" title="E27">A19</font></i></a>, <a class="txt" href="latsubgr.html#E22:31_1"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="group_2.html#T61" title="GROUP_2:th.61">GROUP_2:61</a></i>;<br/>
<a NAME="E26:31_1"/>
 <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c13">B2</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E25:31_1"><i><font color="Green" title="E29">A28</font></i></a>, <a class="ref" href="latsubgr.html#T3" target="_self" title="LATSUBGR:th.3">Th3</a></i>;<br/>
<a NAME="E27:31_1"/><b>then </b><i><font color="Green" title="E30">A29</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="txt" href="latsubgr.html#E1:31_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="latsubgr.html#E7:31_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="latsubgr.html#E16:31_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="txt" href="latsubgr.html#E25:31_1"><i><font color="Green" title="E29">A28</font></i></a>, <a class="ref" href="latsubgr.html#T3" target="_self" title="LATSUBGR:th.3">Th3</a></i>;<br/>
<a NAME="E28:31_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">f</font> )
 
<b>by </b><i><a class="txt" href="latsubgr.html#E8:31_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a></i>;<br/>
<a NAME="E29:31_1"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> )
 
<b>by </b><i><a class="txt" href="latsubgr.html#E1:31"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="latsubgr.html#E27:31_1"><i><font color="Green" title="E30">A29</font></i></a>, <a class="ref" href="funct_1.html#T157" title="FUNCT_1:th.157">FUNCT_1:157</a></i>;<br/>
<a NAME="E30:31_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E31:31_1"/>
<font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font>
 <b>by </b><i><a class="ref" href="group_2.html#T68" title="GROUP_2:th.68">GROUP_2:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
 <a href="latsubgr.html#K3" title="LATSUBGR:func.3">FuncLatt</a> <font color="Maroon" title="c3">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div>
