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

<a NAME="E1:47"/><i><font color="Green">E25</font></i>: 
 <a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a>  <a href="realset1.html#K1">||</a> <a href="nat_lat.html#K7" target="_self">NATPLUS</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_1"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:47_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:47_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E4:47_1_1"/>
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="txt" href="nat_lat.html#E2:47_1_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  holds <br/><a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K11" target="_self">lcmlatplus</a> 
 ;<br/>

<a NAME="E2:47_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:47_1_2"/><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E5:47_1_2"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="nat_lat.html#K7" target="_self">NATPLUS</a>  &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="nat_lat.html#K7" target="_self">NATPLUS</a>  )
 
<b>by </b><i>, <a class="txt" href="nat_lat.html#E2:47_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">y1</font>, <font color="Maroon">k</font> = <font color="Maroon">y2</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:47_1_2"/><i><font color="Green">E71</font></i>: 
<a href="nat_lat.html#K2" target="_self">lcmlat</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K8">lcm</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">y1</font>, <font color="Maroon">k</font> = <font color="Maroon">y2</font> as   <a href="nat_lat.html#NM2">NatPlus</a> <b>by </b><i>, <a class="txt" href="nat_lat.html#E2:47_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:47_1_2"/>
<a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="nat_lat.html#E4:47_1_2"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E10:47_1_2"/>
<a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="nat_lat.html#E2:47_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_1"/>
 <a href="nat_lat.html#K11" target="_self">lcmlatplus</a>  <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K2" target="_self">lcmlat</a>  <a href="realset1.html#K1">||</a> <a href="nat_lat.html#K7" target="_self">NATPLUS</a> 
 <b>by </b><i><a class="ref" href="nat_lat.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47"/>
 <a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K1" target="_self">hcflat</a>  <a href="realset1.html#K1">||</a> <a href="nat_lat.html#K7" target="_self">NATPLUS</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><div class="add">

<a NAME="E1:47_2"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K1" target="_self">hcflat</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_2_1"/><i><font color="Green">E73</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:47_2_1"/><i><font color="Green">E90</font></i>: 
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K1" target="_self">hcflat</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:47_2_1"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E4:47_2_1"/>
 <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K1" target="_self">hcflat</a> </span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:47_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K10" target="_self">hcflatplus</a>  holds <br/><a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K1" target="_self">hcflat</a>  <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:47_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <a href="nat_lat.html#K10" target="_self">hcflatplus</a> 
 ;<br/>

<a NAME="E2:47_2_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="nat_lat.html#K7" target="_self">NATPLUS</a> ,<a href="nat_lat.html#K7" target="_self">NATPLUS</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:47_2_2"/><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E5:47_2_2"/>
( <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a>  <a href="nat_lat.html#K7" target="_self">NATPLUS</a>  &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="nat_lat.html#K7" target="_self">NATPLUS</a>  )
 
<b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">n</font> = <font color="Maroon">y1</font>, <font color="Maroon">k</font> = <font color="Maroon">y2</font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:47_2_2"/><i><font color="Green">E93</font></i>: 
<a href="nat_lat.html#K1" target="_self">hcflat</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">y1</font>, <font color="Maroon">k</font> = <font color="Maroon">y2</font> as   <a href="nat_lat.html#NM2">NatPlus</a> <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:47_2_2"/>
<a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">n</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K1" target="_self">hcflat</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">n</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:47_2_2"/>
<a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K1" target="_self">hcflat</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47_2"/>
 <a href="nat_lat.html#K10" target="_self">hcflatplus</a>  <a href="hidden.html#R1">=</a> <a href="nat_lat.html#K1" target="_self">hcflat</a>  <a href="realset1.html#K1">||</a> <a href="nat_lat.html#K7" target="_self">NATPLUS</a> 
 <b>by </b><i><a class="ref" href="nat_lat.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:47"/>
 <a href="nat_lat.html#K12" target="_self">NatPlus_Lattice</a>  is    <a href="nat_lat.html#M2" target="_self">SubLattice</a> of  <a href="nat_lat.html#K6" target="_self">Nat_Lattice</a> 
 <b>by </b><i>, </i>;<br/>


</div>
