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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">E</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font>;<br/>
<b>consider </b><font color="Maroon">E0</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:10_1_1"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:10_1_1"/><i><font color="Green">E28</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:10_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">E</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a> holds <br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_1"/>
<font color="Maroon">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 ;<br/>

<a NAME="E2:10_1_1_1"/><b>then </b>
<font color="Maroon">E</font> is <a href="intpro_1.html#V1" target="_self">with_FALSUM</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:10_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">E0</font> = <font color="Maroon">E0</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">E0</font>
;<br/>

<a NAME="E6:10_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">E0</font> <a href="tarski.html#R1">c=</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:10_1_1_2"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:10_1_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">0</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:10_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">E0</font> is <a href="intpro_1.html#V1" target="_self">with_FALSUM</a>
 
<b>by </b><i/>;<br/>
<a NAME="E9:10_1_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Olive">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">p</font> = 5 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<a NAME="E2:10_1_1_3"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E3:10_1_1_3"/>
for <font color="Olive">E</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a> holds <br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_3_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_3_1"/>
<font color="Maroon">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 ;<br/>

<a NAME="E2:10_1_1_3_1"/><b>then </b>
<font color="Maroon">E</font> is <a href="intpro_1.html#V5" target="_self">with_int_propositional_variables</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:10_1_1_3_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10_1_1_3"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default">5 <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:10_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">E0</font> is <a href="intpro_1.html#V5" target="_self">with_int_propositional_variables</a>
 
<b>by </b><i/>;<br/>
<a NAME="E11:10_1_1"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1_4"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> )
 ;<br/>

<a NAME="E2:10_1_1_4"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:10_1_1_4"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E5:10_1_1_4"/>
for <font color="Olive">E</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_4_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 ;<br/>

<a NAME="E2:10_1_1_4_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V2" target="_self">with_int_implication</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1_1_4_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 
<b>by </b><i>, , <a class="ref" href="intpro_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_1_4_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="intpro_1.html#D5" target="_self">Def5</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 <b>by </b><i>, <a class="ref" href="intpro_1.html#D3" target="_self">Def3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:10_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">E0</font> is <a href="intpro_1.html#V2" target="_self">with_int_implication</a>
 
<b>by </b><i/>;<br/>
<a NAME="E13:10_1_1"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1_5"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> )
 ;<br/>

<a NAME="E2:10_1_1_5"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  )
 
<b>by </b><i>, <a class="ref" href="intpro_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:10_1_1_5"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E5:10_1_1_5"/>
for <font color="Olive">E</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_5_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 ;<br/>

<a NAME="E2:10_1_1_5_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V3" target="_self">with_int_conjunction</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1_1_5_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 
<b>by </b><i>, <a class="ref" href="intpro_1.html#D3" target="_self">Def3</a>, <a class="ref" href="intpro_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_1_5_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1_1_5"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">2</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 <b>by </b><i>, <a class="ref" href="intpro_1.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:10_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">E0</font> is <a href="intpro_1.html#V3" target="_self">with_int_conjunction</a>
 
<b>by </b><i/>;<br/>
<a NAME="E15:10_1_1"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1_6"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> )
 ;<br/>

<a NAME="E2:10_1_1_6"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a>  )
 
<b>by </b><i>, <a class="ref" href="intpro_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:10_1_1_6"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p'</font></span>)</span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E5:10_1_1_6"/>
for <font color="Olive">E</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_6_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 ;<br/>

<a NAME="E2:10_1_1_6_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V4" target="_self">with_int_disjunction</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1_1_6_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 
<b>by </b><i>, <a class="ref" href="intpro_1.html#D4" target="_self">Def4</a>, </i>;<br/>
<b>hence </b><a NAME="E4:10_1_1_6_1"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="intpro_1.html#D6" target="_self">Def6</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1_1_6"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default">3</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 <b>by </b><i>, <a class="ref" href="intpro_1.html#D5" target="_self">Def5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:10_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">E0</font> is <a href="intpro_1.html#V4" target="_self">with_int_disjunction</a>
 
<b>by </b><i/>;<br/>
<a NAME="E17:10_1_1"/>
for <font color="Olive">p</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font> holds <br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">6</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>assume </b><a NAME="E1:10_1_1_7"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 ;<br/>

<a NAME="E2:10_1_1_7"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i>, <a class="ref" href="intpro_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E4:10_1_1_7"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">6</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <a href="numbers.html#K5">NAT</a>  <a href="finseq_1.html#K13">*</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E5:10_1_1_7"/>
for <font color="Olive">E</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a> holds <br/><span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">6</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">E</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1_7_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:10_1_1_7_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 ;<br/>

<a NAME="E2:10_1_1_7_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V6" target="_self">with_modal_operator</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1_1_7_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="ref" href="intpro_1.html#D5" target="_self">Def5</a>, <a class="ref" href="intpro_1.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1_1_7_1"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">6</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="intpro_1.html#D7" target="_self">Def7</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:10_1_1_7"/>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default">6</span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:10_1_1"/><b>then </b>
<font color="Maroon">E0</font> is <a href="intpro_1.html#V6" target="_self">with_modal_operator</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E19:10_1_1"/>
<font color="Maroon">E0</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 <b>by </b><i><a class="ref" href="intpro_1.html#D1" target="_self">Def1</a>, <a class="ref" href="intpro_1.html#D2" target="_self">Def2</a>, , , <a class="ref" href="intpro_1.html#D3" target="_self">Def3</a>, <a class="ref" href="intpro_1.html#D4" target="_self">Def4</a>, </i>;<br/>

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

<b>assume </b><a NAME="E20:10_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">E</font> is <a href="intpro_1.html#V7" target="_self">MC-closed</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E21:10_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E0</font>
 ;<br/>

<b>hence </b><a NAME="E22:10_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="intpro_1.html#D5" target="_self">Def5</a></i>;<br/>


</div>
