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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">EqR1</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">EqR2</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">EqR3</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/>







<a NAME="E1:1"/>
for <font color="Olive">EqR4</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>  st <font color="Olive">EqR4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> holds <br/><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Olive">EqR4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">EqR4</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E2:1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E4:1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">EqR1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR2</font> &amp; <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>assume </b><a NAME="E5:1_1"/>
<font color="Maroon">EqR4</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font>
 ;<br/>

<a NAME="E6:1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E7:1_1"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:1_1"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">EqR1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:1_1"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E11:1_1"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E12:1_1"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:1_1"/>
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font>
 
;<br/>
<a NAME="E3:1"/>
for <font color="Olive">EqR4</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>  st <font color="Olive">EqR4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Olive">EqR4</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">EqR4</font> be   <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E2:1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">EqR1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">EqR2</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E4:1_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR2</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font> &amp; <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR2</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>assume </b><a NAME="E5:1_2"/>
<font color="Maroon">EqR4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 ;<br/>

<a NAME="E6:1_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E7:1_2"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E8:1_2"/><b>then </b>
<font color="Maroon">EqR2</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">EqR1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font> &amp; <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E10:1_2"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E11:1_2"/><b>then </b>
<font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>
<a NAME="E12:1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K3">\/</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E13:1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR4</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D3">EQREL_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E5:1"/>
<span class="p1">(<span class="default"><font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR2</font></span>)</span> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">EqR1</font> <a href="eqrel_1.html#K5">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">EqR2</font> <a href="eqrel_1.html#K5">"\/"</a> <font color="Maroon">EqR3</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
