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



<a NAME="E1:67"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="classes1.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E2:67"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a> &amp;  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 
<b>by </b><i><a class="ref" href="classes1.html#T58" target="_self">Th58</a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E3:67"/><b>then </b>
(  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="ordinal1.html#V1">epsilon-transitive</a> )
 
<b>by </b><i><a class="ref" href="classes1.html#T55" target="_self">Th55</a>, <a class="ref" href="classes1.html#T65" target="_self">Th65</a></i>;<br/>
<b>hence </b><a NAME="E4:67"/>
 <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="classes1.html#T62" target="_self">Th62</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="E5:67"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:67"/><b>then </b>
(  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="classes1.html#T65" target="_self">Th65</a></i>;<br/>
<b>hence </b><a NAME="E7:67"/>
<span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="classes1.html#K5" target="_self">the_transitive-closure_of</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>2</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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