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<b>let </b><font color="Maroon">IT</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>, <font color="Maroon">PA</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>, <font color="Maroon">PB</font> be    <a href="eqrel_1.html#M1">a_partition</a> of <font color="Maroon">Y</font>;<br/>

<div><i><font color="Green">E91</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1_3_1"/>
<font color="Maroon">a</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 ;<br/><a NAME="E2:22_1_3_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> &amp; <font color="Maroon">a</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> holds <br/><font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_1_3_1"/>
<font color="Maroon">a</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PB</font>,<font color="Maroon">PA</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E92</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1_3_2"/>
<font color="Maroon">a</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PB</font>,<font color="Maroon">PA</font>
 ;<br/><a NAME="E2:22_1_3_2"/><b>then </b>
( <font color="Maroon">a</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Maroon">a</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> &amp; ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PA</font> &amp; <font color="Olive">d</font> <a href="partit1.html#R1" target="_self">is_a_dependent_set_of</a> <font color="Maroon">PB</font> holds <br/><font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:22_1_3_2"/>
<font color="Maroon">a</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:22_1_3"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> iff <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> ) ) holds <br/>for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Olive">a</font> iff <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PB</font>,<font color="Maroon">PA</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_3_3"><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>assume </b><a NAME="E1:22_1_3_3"/><i><font color="Green">E93</font></i>: 
for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> iff <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> )
 ;<br/>

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

<a NAME="E2:22_1_3_3"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> iff <font color="Maroon">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:22_1_3_3"/>
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> implies <font color="Maroon">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PB</font>,<font color="Maroon">PA</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E4:22_1_3_3"/>
<font color="Maroon">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PB</font>,<font color="Maroon">PA</font>
 ;<br/>

<b>hence </b><a NAME="E5:22_1_3_3"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22_1_3"/>
( ( for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> iff <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PA</font>,<font color="Maroon">PB</font> ) ) implies for <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IT</font> iff <font color="Olive">d</font> <a href="partit1.html#R2" target="_self">is_min_depend</a> <font color="Maroon">PB</font>,<font color="Maroon">PA</font> ) )
 ;<br/>


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