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<b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>, <font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> iff  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) )
 <b>and </b><br/><a NAME="E2:22_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> iff  ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> ) )
 ;<br/>

<a NAME="E3:22_1_2"/>
for <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> iff <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:22_1_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 ;<br/>

<a NAME="E2:22_1_2_1_1"/><b>then </b>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:22_1_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:22_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font>
 ;<br/>

<a NAME="E3:22_1_2_1"/><b>then </b>
 ex <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp;  ex <font color="Olive">C</font> being   <a href="measure3.html#M1" target="_self">thin</a> of <font color="Maroon">M</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">C</font> )
 
<b>by </b><i><a class="txt" href="measure3.html#E1"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_2_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_1_2"/>
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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