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

<a NAME="E1:44"/><i><font color="Green" title="E20">A1</font></i>: 
 <a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_1"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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="E2:44_1_1"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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="E3:44_1_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E4:44_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="nat_lat.html#E1:44_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="nat_lat.html#E2:44_1_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:44_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  holds <br/><a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:44_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a> 
 ;<br/>

<a NAME="E2:44_1_2"/><b>then </b><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:44_1_2"/><i><font color="Green" title="E23">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102" title="ZFMISC_1:th.102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E5:44_1_2"/>
( <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a>  &amp; <font color="Maroon" title="c3">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a>  )
 
<b>by </b><i><a class="txt" href="nat_lat.html#E2:44_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="nat_lat.html#E4:44_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c3">y2</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:44_1_2"/><i><font color="Green" title="E24">A7</font></i>: 
<a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_d.html#K6" title="NAT_D:func.6">lcm</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="ref" href="nat_lat.html#D4" target="_self" title="NAT_LAT:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c3">y2</font> as   <a href="nat_lat.html#NM2" title="NAT_LAT:NM.2">NatPlus</a> <b>by </b><i><a class="txt" href="nat_lat.html#E2:44_1_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="nat_lat.html#E4:44_1_2"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:44_1_2"/>
<a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="nat_lat.html#E7:44_1_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="nat_lat.html#D13" target="_self" title="NAT_LAT:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E10:44_1_2"/>
<a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="nat_lat.html#E4:44_1_2"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:44_1"/>
 <a href="nat_lat.html#K10" title="NAT_LAT:func.10">lcmlatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K2" title="NAT_LAT:func.2">lcmlat</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> 
 <b>by </b><i><a class="txt" href="nat_lat.html#E1:44_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:44"/>
 <a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<a NAME="E1:44_2"/><i><font color="Green" title="E21">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:44_2_1"/><i><font color="Green" title="E22">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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="E2:44_2_1"/><i><font color="Green" title="E23">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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="E3:44_2_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E4:44_2_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="nat_lat.html#E1:44_2_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="nat_lat.html#E2:44_2_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:44_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  holds <br/><a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:44_2_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a> 
 ;<br/>

<a NAME="E2:44_2_2"/><b>then </b><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> ,<a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:44_2_2"/><i><font color="Green" title="E23">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">y1</font>,<font color="Maroon" title="c3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102" title="ZFMISC_1:th.102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E5:44_2_2"/>
( <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a>  &amp; <font color="Maroon" title="c3">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a>  )
 
<b>by </b><i><a class="txt" href="nat_lat.html#E2:44_2_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="nat_lat.html#E4:44_2_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c5">k</font> = <font color="Maroon" title="c3">y2</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:44_2_2"/><i><font color="Green" title="E24">A13</font></i>: 
<a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c5">k</font>
 
<b>by </b><i><a class="ref" href="nat_lat.html#D3" target="_self" title="NAT_LAT:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c3">y2</font> as   <a href="nat_lat.html#NM2" title="NAT_LAT:NM.2">NatPlus</a> <b>by </b><i><a class="txt" href="nat_lat.html#E2:44_2_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="nat_lat.html#E4:44_2_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:44_2_2"/>
<a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">n</font>,<font color="Maroon" title="c7">k</font>
 
<b>by </b><i><a class="txt" href="nat_lat.html#E7:44_2_2"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="nat_lat.html#D12" target="_self" title="NAT_LAT:def.12">Def12</a></i>;<br/>
<b>hence </b><a NAME="E10:44_2_2"/>
<a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="nat_lat.html#E4:44_2_2"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:44_2"/>
 <a href="nat_lat.html#K9" title="NAT_LAT:func.9">hcflatplus</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="nat_lat.html#K1" title="NAT_LAT:func.1">hcflat</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="nat_lat.html#K6" title="NAT_LAT:func.6">NATPLUS</a> 
 <b>by </b><i><a class="txt" href="nat_lat.html#E1:44_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:44"/>
 <a href="nat_lat.html#K11" title="NAT_LAT:func.11">NatPlus_Lattice</a>  is    <a href="nat_lat.html#M2" title="NAT_LAT:mode.2">SubLattice</a> of  <a href="nat_lat.html#K3" title="NAT_LAT:func.3">Nat_Lattice</a> 
 <b>by </b><i><a class="txt" href="nat_lat.html#E1:44"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="nat_lat.html#D16" target="_self" title="NAT_LAT:def.16">Def16</a></i>;<br/>


</div>
