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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8" target="_self">Normal_forms_on</a> <font color="Maroon">c<sub>1</sub></font>;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="normform.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E4:61_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="normform.html#E3:61_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:61_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="normform.html#E3:61_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="normform.html#T56" target="_self">Th56</a></i>;<br/><b>hence </b><a NAME="E6:61_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:61"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="normform.html#E77"><i><font color="Green">Lemma32</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<a NAME="E3:61"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:61"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="normform.html#T73" target="_self">Th73</a></i>;<br/>
<b>hence </b><a NAME="E5:61"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="setwiseo.html#K5">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normform.html#K10" target="_self">^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>


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