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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>thus </b><a NAME="E1:87"/>
( <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> implies ( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_1"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 ;<br/>

<b>assume </b><a NAME="E2:87_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:87_1"/><b>then </b>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or ( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
;<br/>
<a NAME="E4:87_1"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> or <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T84" target="_self">Th84</a>, <a class="ref" href="finseq_3.html#T85" target="_self">Th85</a>, <a class="ref" href="finseq_3.html#T87" target="_self">Th87</a></i>;<br/>
<b>hence </b><a NAME="E5:87_1"/>
contradiction
 <b>by </b><i><a class="txt" href="finseq_3.html#E1:87_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_3.html#T38" target="_self">Th38</a>, <a class="ref" href="finseq_3.html#T40" target="_self">Th40</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:87"/>
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:87"/><b>then </b>
( <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K9">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="finseq_3.html#E87"><i><font color="Green">Lemma47</font></i></a>, <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:87"/>
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="finseq_3.html#K1" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="finseq_3.html#E92"><i><font color="Green">Lemma52</font></i></a></i>;<br/>


</div>
