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<span class="kw">let </span><font color="Maroon" title="c3">A1</font>, <font color="Maroon" title="c4">A2</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A1</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> ) ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">A2</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> ) ) implies <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> ) &amp; (  not <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%I</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%I</a> <font color="Maroon" title="c1">Y</font> implies <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> ) )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A1</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> ) ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">A2</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> ) ) implies <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:7_1_2_1"/>
<font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:7_1_2_1"/><span class="lab"><font color="Green" title="E4">A26</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A1</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> )
 <span class="kw">and </span><br/><a NAME="E3:7_1_2_1"/><span class="lab"><font color="Green" title="E5">A27</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">A2</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font></span><br/><a NAME="E4:7_1_2_1"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A1</font> iff <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:7_1_2_1"><span class="lab"><font color="Green" title="E5">A27</font></span></a>, <a class="txt" href="#E2:7_1_2_1"><span class="lab"><font color="Green" title="E4">A26</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:7_1_2_1"/>
<font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:7_1_2"/>
( ( <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">A1</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> ) ) &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">A2</font> iff <font color="Olive" title="b1">x</font> <a href="bvfunc_2.html#R1" title="BVFUNC_2:pred.1">is_upper_min_depend_of</a> <font color="Maroon" title="c2">G</font> ) ) implies <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> ) &amp; (  not <font color="Maroon" title="c2">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%I</a> <font color="Maroon" title="c1">Y</font> &amp; <font color="Maroon" title="c4">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partit1.html#K6" title="PARTIT1:func.6">%I</a> <font color="Maroon" title="c1">Y</font> implies <font color="Maroon" title="c3">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A2</font> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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