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

<b>set </b><font color="Maroon">Id</font> = <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">I</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> ;<br/>
<a NAME="E1:75_3_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:75_3_1"/><b>then </b><i><font color="Green">E152</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> 
 
;<br/>
<div><i><font color="Green">E153</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_1_1"><b>now </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="E1:75_3_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">X'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:75_3_1_1"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">X'</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> )
 ;<br/><b>thus </b><a NAME="E4:75_3_1_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E4:75_3_1"/><b>then </b><i><font color="Green">E155</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:75_3_1"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i>, , <a class="ref" href="ideal_1.html#D3" target="_self">Def3</a>, <a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E8:75_3_1"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">I</font> is <a href="ideal_1.html#V1" target="_self">add-closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D1" target="_self">IDEAL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:75_3_1_2"/><i><font color="Green">E158</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_1_2_1"><b>now </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="E1:75_3_1_2_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> 
 ;<br/><a NAME="E2:75_3_1_2_1"/><b>then </b><i><font color="Green">E160</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E3:75_3_1_2_1"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:75"><i><font color="Green">E41</font></i></a>, , <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">X'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:75_3_1_2_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">X'</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><a NAME="E6:75_3_1_2_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E7:75_3_1_2_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:75_3_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:75_3_1_2"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:75_3_1"/>
<font color="Maroon">I</font> is <a href="ideal_1.html#V3" target="_self">right-ideal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D3" target="_self">IDEAL_1:def 3</a><br/>

<b>assume </b><a NAME="E1:75_3_1_3"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_1_3_1"><b>now </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="E1:75_3_1_3_1"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> 
 ;<br/><a NAME="E2:75_3_1_3_1"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:75"><i><font color="Green">E41</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">X'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:75_3_1_3_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X'</font> &amp; <font color="Maroon">X'</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><a NAME="E5:75_3_1_3_1"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E6:75_3_1_3_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:75_3_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>
<b>hence </b><a NAME="E4:75_3_1_3"/>
<font color="Maroon">x</font> <a href="group_1.html#K1">*</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">I</font> = <font color="Maroon">I</font> as   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">I</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_3_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">X</font> be   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:75_3_1_4"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:75_3_1_4"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">I</font> where <font color="Olive">I</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> &amp; <font color="Olive">I</font> is   <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font> ) </span>}</span> 
 ;<br/><b>hence </b><a NAME="E3:75_3_1_4"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:75_3_1"/>
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">I</font> being  <a href="ideal_1.html#NM2">RightIdeal</a> of <font color="Maroon">L</font>  st <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> holds <br/><font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> ) )
 <b>by </b><i>, <a class="ref" href="ideal_1.html#D2" target="_self">Def2</a>, <a class="ref" href="setfam_1.html#T6">SETFAM_1:6</a></i>;<br/>


</div>
