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<span class="kw">let </span><font color="Maroon" title="c4">IT</font>, <font color="Maroon" title="c5">PA</font>, <font color="Maroon" title="c6">PB</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> ) ) implies for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E16">A61</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> implies <font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1_3_1"/><span class="lab"><font color="Green" title="E17">A62</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font></span><br/><a NAME="E2:24_1_3_1"/><span class="lab"><font color="Green" title="E18">A63</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c6">PB</font> &amp; <font color="Maroon" title="c7">a</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c5">PA</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_3_1"><span class="lab"><font color="Green" title="E17">A62</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/><a NAME="E3:24_1_3_1"/><span class="lab"><font color="Green" title="E19">A64</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c6">PB</font> &amp; <font color="Olive" title="b1">d</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c5">PA</font> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_3_1"><span class="lab"><font color="Green" title="E17">A62</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/><span class="kw">thus </span><a NAME="E4:24_1_3_1"/>
<font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_3_1"><span class="lab"><font color="Green" title="E18">A63</font></span></a>, <a class="txt" href="#E3:24_1_3_1"><span class="lab"><font color="Green" title="E19">A64</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E17">A65</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> implies <font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1_3_2"/><span class="lab"><font color="Green" title="E18">A66</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font></span><br/><a NAME="E2:24_1_3_2"/><span class="lab"><font color="Green" title="E19">A67</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c5">PA</font> &amp; <font color="Maroon" title="c7">a</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c6">PB</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_3_2"><span class="lab"><font color="Green" title="E18">A66</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/><a NAME="E3:24_1_3_2"/><span class="lab"><font color="Green" title="E20">A68</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Olive" title="b1">d</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c5">PA</font> &amp; <font color="Olive" title="b1">d</font> <a href="partit1.html#R1" title="PARTIT1:pred.1">is_a_dependent_set_of</a> <font color="Maroon" title="c6">PB</font> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_3_2"><span class="lab"><font color="Green" title="E18">A66</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>;<br/><span class="kw">thus </span><a NAME="E4:24_1_3_2"/>
<font color="Maroon" title="c7">a</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_3_2"><span class="lab"><font color="Green" title="E19">A67</font></span></a>, <a class="txt" href="#E3:24_1_3_2"><span class="lab"><font color="Green" title="E20">A68</font></span></a>, <a class="ref" href="partit1.html#D2" target="_self" title="PARTIT1:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:24_1_3"/><span class="lab"><font color="Green" title="E18">A69</font></span>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st ( for <font color="Olive" title="b2">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> iff <font color="Olive" title="b2">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> ) ) holds <br/>for <font color="Olive" title="b2">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font> iff <font color="Olive" title="b2">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> ) ) implies for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_3_3"/><span class="lab"><font color="Green" title="E19">A70</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> iff <font color="Maroon" title="c8">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> )</span><br/>

<a NAME="E2:24_1_3_3"/><span class="lab"><font color="Green" title="E20">A71</font></span>: 
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> iff <font color="Maroon" title="c8">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_3_3"><span class="lab"><font color="Green" title="E19">A70</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:24_1_3_3"/>
( <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> implies <font color="Maroon" title="c8">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_3"><span class="lab"><font color="Green" title="E16">A61</font></span></a>, <a class="txt" href="#E2:24_1_3_3"><span class="lab"><font color="Green" title="E20">A71</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> implies <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:24_1_3_3"/><span class="lab"><font color="Green" title="E21">A72</font></span>: 
<font color="Maroon" title="c8">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font></span><br/>

<span class="kw">thus </span><a NAME="E5:24_1_3_3"/>
<font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_3"><span class="lab"><font color="Green" title="E17">A65</font></span></a>, <a class="txt" href="#E2:24_1_3_3"><span class="lab"><font color="Green" title="E20">A71</font></span></a>, <a class="txt" href="#E4:24_1_3_3"><span class="lab"><font color="Green" title="E21">A72</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E4:24_1_3"/>
( ( for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c5">PA</font>,<font color="Maroon" title="c6">PB</font> ) ) implies for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff <font color="Olive" title="b1">d</font> <a href="partit1.html#R2" title="PARTIT1:pred.2">is_min_depend</a> <font color="Maroon" title="c6">PB</font>,<font color="Maroon" title="c5">PA</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_3"><span class="lab"><font color="Green" title="E18">A69</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
