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<div class="add">

<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">G</font> is <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:15"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c4">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1"/>
( <font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c3">P</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="c4">Q</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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_1"><b>suppose </b></a><a NAME="E1:15_1_1"/>
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:15_1_1"/><b>then </b><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 ;<br/><b>then </b><i><font color="Green" title="E15">A5</font></i>: <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<b>by </b><i><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></i>
;<br/><span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="partit_2.html#E2:15_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<b>by </b><i><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></i>
;<br/><b>hence </b><a NAME="E5:15_1_1"/>
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_1"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_2"><b>suppose </b></a><a NAME="E1:15_1_2"/>
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:15_1_2"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>
 ;<br/><b>then </b><i><font color="Green" title="E15">A7</font></i>: <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<b>by </b><i><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></i>
;<br/><span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="partit1.html#K6" title="PARTIT1:func.6">%O</a> <font color="Maroon" title="c1">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="partit_2.html#E2:15_1_2"><i><font color="Green" title="E14">A6</font></i></a>, <a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></i>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<b>by </b><i><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></i>
;<br/><b>hence </b><a NAME="E5:15_1_2"/>
<span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_2"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3"><b>suppose </b></a><a NAME="E1:15_1_3"/><i><font color="Green" title="E14">A8</font></i>: 
( <font color="Maroon" title="c3">P</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="c4">Q</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>  )
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="relset_1.html#D2" title="RELSET_1:def.2">RELSET_1:def 2</a><br/><b>assume </b><a NAME="E2:15_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E4:15_1_3"/><i><font color="Green" title="E15">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span>
 <b>and </b><br/><a NAME="E5:15_1_3"/><i><font color="Green" title="E16">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T51" title="RELSET_1:th.51">RELSET_1:51</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E7:15_1_3"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font>
 <b>and </b><br/><a NAME="E8:15_1_3"/><i><font color="Green" title="E18">A12</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <b>and </b><br/><a NAME="E9:15_1_3"/><i><font color="Green" title="E19">A13</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E4:15_1_3"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E11:15_1_3"/><i><font color="Green" title="E20">A14</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font>
 <b>and </b><br/><a NAME="E12:15_1_3"/><i><font color="Green" title="E21">A15</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>and </b><br/><a NAME="E13:15_1_3"/><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">B</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E5:15_1_3"><i><font color="Green" title="E16">A10</font></i></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">hP</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c11">FP</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E15:15_1_3"/><i><font color="Green" title="E23">A17</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">hP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font>
 <b>and </b><br/><a NAME="E16:15_1_3"/><i><font color="Green" title="E24">A18</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">hP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">FP</font>
 <b>and </b><br/><a NAME="E17:15_1_3"/><i><font color="Green" title="E25">A19</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/><font color="Maroon" title="c10">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d</font>
 <b>and </b><br/><a NAME="E18:15_1_3"/><i><font color="Green" title="E26">A20</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c11">FP</font>
 <b>and </b><br/><a NAME="E19:15_1_3"/>
<font color="Maroon" title="c8">A</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> 
 <b>by </b><i><a class="txt" href="partit_2.html#E7:15_1_3"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">hQ</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c13">FQ</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E21:15_1_3"/><i><font color="Green" title="E27">A21</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">hQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font>
 <b>and </b><br/><a NAME="E22:15_1_3"/><i><font color="Green" title="E28">A22</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">hQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">FQ</font>
 <b>and </b><br/><a NAME="E23:15_1_3"/><i><font color="Green" title="E29">A23</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> holds <br/><font color="Maroon" title="c12">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d</font>
 <b>and </b><br/><a NAME="E24:15_1_3"/><i><font color="Green" title="E30">A24</font></i>: 
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c13">FQ</font>
 <b>and </b><br/><a NAME="E25:15_1_3"/>
<font color="Maroon" title="c9">B</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> 
 <b>by </b><i><a class="txt" href="partit_2.html#E11:15_1_3"><i><font color="Green" title="E20">A14</font></i></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">P</font> = <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c15">Q</font> = <font color="Maroon" title="c4">Q</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span> <b>by </b><i><a class="txt" href="partit_2.html#E1:15_1_3"><i><font color="Green" title="E14">A8</font></i></a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c14">P</font>) <b>-&gt; </b>   <a href="bvfunc_1.html#M1" title="BVFUNC_1:mode.1">Element</a> of $1 =  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,$1;<br/><b>consider </b><font color="Maroon" title="c16">hP'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c14">P</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E28:15_1_3"/><i><font color="Green" title="E31">A25</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c14">P</font> holds  <font color="Maroon" title="c16">hP'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E29:15_1_3"/><i><font color="Green" title="E32">A26</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> holds <br/><font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_1"/>
<font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c18">d</font> = <font color="Maroon" title="c17">d</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c14">P</font> ;<br/>
<a NAME="E3:15_1_3_1"/>
<font color="Maroon" title="c16">hP'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c18">d</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E28:15_1_3"><i><font color="Green" title="E31">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_3_1"/>
<font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">d</font>
 ;<br/>


</div><b>end;</b></div><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c15">Q</font>) <b>-&gt; </b>   <a href="bvfunc_1.html#M1" title="BVFUNC_1:mode.1">Element</a> of $1 =  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c5">x</font>,$1;<br/><b>consider </b><font color="Maroon" title="c17">hQ'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">Q</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E31:15_1_3"/><i><font color="Green" title="E33">A27</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c15">Q</font> holds  <font color="Maroon" title="c17">hQ'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">p</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><a NAME="E32:15_1_3"/><i><font color="Green" title="E34">A28</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font> holds <br/><font color="Maroon" title="c17">hQ'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_2"/>
<font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c19">d</font> = <font color="Maroon" title="c18">d</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c15">Q</font> ;<br/>
<a NAME="E3:15_1_3_2"/>
<font color="Maroon" title="c17">hQ'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c19">d</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E31:15_1_3"><i><font color="Green" title="E33">A27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:15_1_3_2"/>
<font color="Maroon" title="c17">hQ'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">d</font>
 ;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c18">FP'</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">hP'</font>, <font color="Maroon" title="c19">FQ'</font> =  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">hQ'</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> ;<br/><b>set </b><font color="Maroon" title="c20">A'</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c18">FP'</font>;<br/><b>set </b><font color="Maroon" title="c21">B'</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c19">FQ'</font>;<br/><a NAME="E34:15_1_3"/><i><font color="Green" title="E35">A29</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">hP'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">P</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E35:15_1_3"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">FP'</font> holds <br/><font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_3"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">FP'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:15_1_3_3"/><i><font color="Green" title="E36">A30</font></i>: 
<font color="Maroon" title="c23">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">hP'</font>
 <b>and </b><br/><a NAME="E4:15_1_3_3"/><i><font color="Green" title="E37">A31</font></i>: 
<font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">b</font> = <font color="Maroon" title="c23">b</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c14">P</font> <b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_3_3"><i><font color="Green" title="E36">A30</font></i></a></i>;<br/>
<a NAME="E6:15_1_3_3"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c24">b</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E28:15_1_3"><i><font color="Green" title="E31">A25</font></i></a>, <a class="txt" href="partit_2.html#E4:15_1_3_3"><i><font color="Green" title="E37">A31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_3_3"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">a</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E36:15_1_3"/><b>then </b><i><font color="Green" title="E36">A32</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c18">FP'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><a NAME="E37:15_1_3"/><i><font color="Green" title="E37">A33</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">hQ'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">Q</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E38:15_1_3"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">FQ'</font> holds <br/><font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_4"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">FQ'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:15_1_3_4"/><i><font color="Green" title="E38">A34</font></i>: 
<font color="Maroon" title="c23">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">hQ'</font>
 <b>and </b><br/><a NAME="E4:15_1_3_4"/><i><font color="Green" title="E39">A35</font></i>: 
<font color="Maroon" title="c17">hQ'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c24">b</font> = <font color="Maroon" title="c23">b</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c15">Q</font> <b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_3_4"><i><font color="Green" title="E38">A34</font></i></a></i>;<br/>
<a NAME="E6:15_1_3_4"/>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c24">b</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E31:15_1_3"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="partit_2.html#E4:15_1_3_4"><i><font color="Green" title="E39">A35</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_3_4"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">a</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E39:15_1_3"/><b>then </b><i><font color="Green" title="E38">A36</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c19">FQ'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">G'</font> = <font color="Maroon" title="c2">G</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span> <b>by </b><i><a class="txt" href="partit_2.html#E2:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/><a NAME="E41:15_1_3"/><i><font color="Green" title="E39">A37</font></i>: 
for <font color="Olive" title="b1">d</font> being   <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c22">G'</font>  holds  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">d</font>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">d</font> be    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c22">G'</font>;<br/>

<a NAME="E1:15_1_3_5"/>
<font color="Maroon" title="c23">d</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c23">d</font>
 
;<br/>
<b>hence </b><a NAME="E2:15_1_3_5"/>
 <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c23">d</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c23">h'</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c22">G'</font>, <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E43:15_1_3"/><i><font color="Green" title="E40">A38</font></i>: 
for <font color="Olive" title="b1">d</font> being   <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c22">G'</font> holds  <font color="Maroon" title="c23">h'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">d</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S11" title="FUNCT_2:sch.11">FUNCT_2:sch 11</a>(<a class="txt" href="partit_2.html#E41:15_1_3"><i><font color="Green" title="E39">A37</font></i></a>);<br/></i><b>set </b><font color="Maroon" title="c24">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font>;<br/><i><font color="Green" title="E41">A39</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">Q</font>
<b>by </b><i><a class="txt" href="partit_2.html#E37:15_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c23">h'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c14">P</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">Q</font>
<b>by </b><i><a class="txt" href="partit_2.html#E34:15_1_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font>
<b>by </b><i><a class="txt" href="partit_2.html#E2:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">G</font>
<b>by </b><i><a class="txt" href="partit_2.html#E3:15"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/><a NAME="E45:15_1_3"/><i><font color="Green" title="E42">A40</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">hQ'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/><a NAME="E46:15_1_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c23">h'</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c16">hP'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></i>;<br/><a NAME="E47:15_1_3"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E48:15_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c25">F</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font> <b>by </b><i><a class="txt" href="partit_2.html#E45:15_1_3"><i><font color="Green" title="E42">A40</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E50:15_1_3"/><i><font color="Green" title="E43">A41</font></i>: 
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_7"/><i><font color="Green" title="E44">A42</font></i>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c27">d'</font> = <font color="Maroon" title="c26">d</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c14">P</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_3_7_1"/>
( <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font> or  not <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_7_1_1"><b>suppose </b></a><a NAME="E1:15_1_3_7_1_1"/><i><font color="Green" title="E45">A43</font></i>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font>
 ;<br/><div class="add"><a NAME="E2:15_1_3_7_1_1"/><b>then </b><i><font color="Green" title="E46">A44</font></i>: 
<font color="Maroon" title="c12">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E23:15_1_3"><i><font color="Green" title="E29">A23</font></i></a></i>;<br/><a NAME="E3:15_1_3_7_1_1"/><i><font color="Green" title="E47">A45</font></i>: 
<font color="Maroon" title="c10">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E17:15_1_3"><i><font color="Green" title="E25">A19</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_7"><i><font color="Green" title="E44">A42</font></i></a></i>;<br/><a NAME="E4:15_1_3_7_1_1"/><i><font color="Green" title="E48">A46</font></i>: 
<font color="Maroon" title="c10">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">FP</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E15:15_1_3"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="partit_2.html#E16:15_1_3"><i><font color="Green" title="E24">A18</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_7"><i><font color="Green" title="E44">A42</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:15_1_3_7_1_1"/><b>then </b><i><font color="Green" title="E49">A47</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E8:15_1_3"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="partit_2.html#E18:15_1_3"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><a NAME="E6:15_1_3_7_1_1"/><i><font color="Green" title="E50">A48</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E9:15_1_3"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="partit_2.html#E18:15_1_3"><i><font color="Green" title="E26">A20</font></i></a>, <a class="txt" href="partit_2.html#E4:15_1_3_7_1_1"><i><font color="Green" title="E48">A46</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><a NAME="E7:15_1_3_7_1_1"/><i><font color="Green" title="E51">A49</font></i>: 
<font color="Maroon" title="c12">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">FQ</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E21:15_1_3"><i><font color="Green" title="E27">A21</font></i></a>, <a class="txt" href="partit_2.html#E22:15_1_3"><i><font color="Green" title="E28">A22</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_7_1_1"><i><font color="Green" title="E45">A43</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:15_1_3_7_1_1"/><b>then </b><i><font color="Green" title="E52">A50</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E13:15_1_3"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="partit_2.html#E24:15_1_3"><i><font color="Green" title="E30">A24</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><a NAME="E9:15_1_3_7_1_1"/><i><font color="Green" title="E53">A51</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E12:15_1_3"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="partit_2.html#E24:15_1_3"><i><font color="Green" title="E30">A24</font></i></a>, <a class="txt" href="partit_2.html#E7:15_1_3_7_1_1"><i><font color="Green" title="E51">A49</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> = 
<font color="Maroon" title="c17">hQ'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E37:15_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_7_1_1"><i><font color="Green" title="E45">A43</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c27">d'</font>
<b>by </b><i><a class="txt" href="partit_2.html#E31:15_1_3"><i><font color="Green" title="E33">A27</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_7_1_1"><i><font color="Green" title="E45">A43</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_3_7_1_1"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="partit_2.html#E5:15_1_3_7_1_1"><i><font color="Green" title="E49">A47</font></i></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c27">d'</font>
<b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_3_7_1_1"><i><font color="Green" title="E47">A45</font></i></a>, <a class="txt" href="partit_2.html#E6:15_1_3_7_1_1"><i><font color="Green" title="E50">A48</font></i></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c12">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E2:15_1_3_7_1_1"><i><font color="Green" title="E46">A44</font></i></a>, <a class="txt" href="partit_2.html#E9:15_1_3_7_1_1"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>
<br/>.= 
 <a href="bvfunc_1.html#K20" title="BVFUNC_1:func.20">EqClass</a> <font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c27">d'</font>
<b>by </b><i><a class="txt" href="partit_2.html#E2:15_1_3_7_1_1"><i><font color="Green" title="E46">A44</font></i></a>, <a class="txt" href="partit_2.html#E8:15_1_3_7_1_1"><i><font color="Green" title="E52">A50</font></i></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E28:15_1_3"><i><font color="Green" title="E31">A25</font></i></a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_7_1_2"><b>suppose </b></a><a NAME="E1:15_1_3_7_1_2"/>
not <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E37:15_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
<b>by </b><i><a class="txt" href="partit_2.html#E34:15_1_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_7"><i><font color="Green" title="E44">A42</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E51:15_1_3"/>
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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_8"/><i><font color="Green" title="E44">A52</font></i>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 ;<br/>

<a NAME="E2:15_1_3_8"/>
<font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="partit_2.html#E3:15"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>then </b><font color="Maroon" title="c2">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">G</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></i>
;<br/>
<a NAME="E4:15_1_3_8"/><b>then </b><i><font color="Green" title="E45">A53</font></i>: 
( <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font></span>)</span> or <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font> )
 
<b>by </b><i><a class="txt" href="partit_2.html#E1:15_1_3_8"><i><font color="Green" title="E44">A52</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_8_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:15_1_3_8_2"/>
( <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font> or <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> or <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="partit_2.html#E4:15_1_3_8"><i><font color="Green" title="E45">A53</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_8_2_1"><b>suppose </b></a><a NAME="E1:15_1_3_8_2_1"/><i><font color="Green" title="E46">A54</font></i>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Q</font>
 ;<br/><div class="add"><a NAME="E2:15_1_3_8_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">hQ'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E37:15_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E3:15_1_3_8_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E32:15_1_3"><i><font color="Green" title="E34">A28</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_8_2_1"><i><font color="Green" title="E46">A54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_8_2_2"><b>suppose </b></a><a NAME="E1:15_1_3_8_2_2"/><i><font color="Green" title="E46">A55</font></i>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font>
 ;<br/><div class="add"><a NAME="E2:15_1_3_8_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E50:15_1_3"><i><font color="Green" title="E43">A41</font></i></a></i>;<br/><b>hence </b><a NAME="E3:15_1_3_8_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E29:15_1_3"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="partit_2.html#E1:15_1_3_8_2_2"><i><font color="Green" title="E46">A55</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="15_1_3_8_2_3"><b>suppose </b></a><a NAME="E1:15_1_3_8_2_3"/><i><font color="Green" title="E46">A56</font></i>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:15_1_3_8_2_3"/><b>then </b><i><font color="Green" title="E47">A57</font></i>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:15_1_3_8_2_3"/><i><font color="Green" title="E48">A58</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">hP'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a></i>;<br/><a NAME="E4:15_1_3_8_2_3"/>
not <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">Q</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E1:15_1_3_8_2_3"><i><font color="Green" title="E46">A56</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:15_1_3_8_2_3"/><b>then </b>
not <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">hP'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span>
 <b>by </b><i><a class="txt" href="partit_2.html#E34:15_1_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="partit_2.html#E37:15_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>;<br/><a NAME="E6:15_1_3_8_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">h'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_3_8_2_3"><i><font color="Green" title="E48">A58</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><b>hence </b><a NAME="E7:15_1_3_8_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">d</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E43:15_1_3"><i><font color="Green" title="E40">A38</font></i></a>, <a class="txt" href="partit_2.html#E2:15_1_3_8_2_3"><i><font color="Green" title="E47">A57</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E52:15_1_3"/><b>then </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c25">F</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> 
 <b>by </b><i><a class="txt" href="partit_2.html#E1:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="partit_2.html#E44:15_1_3"><i><font color="Green" title="E41">A39</font></i></a>, <a class="ref" href="bvfunc_2.html#D5" title="BVFUNC_2:def.5">BVFUNC_2:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c26">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E54:15_1_3"/><i><font color="Green" title="E44">A59</font></i>: 
<font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c25">F</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E55:15_1_3"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">FP'</font> holds <br/><font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_9"/>
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">FP'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c28">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:15_1_3_9"/><i><font color="Green" title="E45">A60</font></i>: 
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">hP'</font>
 <b>and </b><br/><a NAME="E4:15_1_3_9"/><i><font color="Green" title="E46">A61</font></i>: 
<font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:15_1_3_9"/>
<font color="Maroon" title="c16">hP'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">b</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E50:15_1_3"><i><font color="Green" title="E43">A41</font></i></a>, <a class="txt" href="partit_2.html#E3:15_1_3_9"><i><font color="Green" title="E45">A60</font></i></a></i>;<br/>
<a NAME="E6:15_1_3_9"/><b>then </b>
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">F</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E2:15"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="partit_2.html#E34:15_1_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="partit_2.html#E44:15_1_3"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="partit_2.html#E3:15_1_3_9"><i><font color="Green" title="E45">A60</font></i></a>, <a class="txt" href="partit_2.html#E4:15_1_3_9"><i><font color="Green" title="E46">A61</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_3_9"/>
<font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">a</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E54:15_1_3"><i><font color="Green" title="E44">A59</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E56:15_1_3"/><b>then </b><i><font color="Green" title="E45">A62</font></i>: 
<font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c18">FP'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><a NAME="E57:15_1_3"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">FQ'</font> holds <br/><font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_3_10"/>
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">FQ'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c28">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:15_1_3_10"/><i><font color="Green" title="E46">A63</font></i>: 
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">hQ'</font>
 <b>and </b><br/><a NAME="E4:15_1_3_10"/><i><font color="Green" title="E47">A64</font></i>: 
<font color="Maroon" title="c17">hQ'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">b</font> = <font color="Maroon" title="c28">b</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c15">Q</font> <b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_3_10"><i><font color="Green" title="E46">A63</font></i></a></i>;<br/>
<a NAME="E6:15_1_3_10"/>
<font color="Maroon" title="c17">hQ'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">h'</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">hP'</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hQ'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">b</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:15_1_3_10"><i><font color="Green" title="E46">A63</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/>
<a NAME="E7:15_1_3_10"/><b>then </b>
<font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">F</font>
 
<b>by </b><i><a class="txt" href="partit_2.html#E3:15"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="partit_2.html#E37:15_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="txt" href="partit_2.html#E44:15_1_3"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="partit_2.html#E3:15_1_3_10"><i><font color="Green" title="E46">A63</font></i></a>, <a class="txt" href="partit_2.html#E4:15_1_3_10"><i><font color="Green" title="E47">A64</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:15_1_3_10"/>
<font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">a</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E54:15_1_3"><i><font color="Green" title="E44">A59</font></i></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div><a NAME="E58:15_1_3"/><b>then </b><i><font color="Green" title="E46">A65</font></i>: 
<font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c19">FQ'</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/><a NAME="E59:15_1_3"/><b>then </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c19">FQ'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c15">Q</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E32:15_1_3"><i><font color="Green" title="E34">A28</font></i></a>, <a class="txt" href="partit_2.html#E37:15_1_3"><i><font color="Green" title="E37">A33</font></i></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></i>;<br/><a NAME="E60:15_1_3"/><b>then </b><i><font color="Green" title="E47">A66</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c26">t</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c15">Q</font></span>)</span>
 <b>by </b><i><a class="txt" href="partit_2.html#E39:15_1_3"><i><font color="Green" title="E38">A36</font></i></a>, <a class="txt" href="partit_2.html#E58:15_1_3"><i><font color="Green" title="E46">A65</font></i></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></i>;<br/><a NAME="E61:15_1_3"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c18">FP'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c14">P</font>
 <b>by </b><i><a class="txt" href="partit_2.html#E29:15_1_3"><i><font color="Green" title="E32">A26</font></i></a>, <a class="txt" href="partit_2.html#E34:15_1_3"><i><font color="Green" title="E35">A29</font></i></a>, <a class="txt" href="partit_2.html#E56:15_1_3"><i><font color="Green" title="E45">A62</font></i></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></i>;<br/><a NAME="E62:15_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">t</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p1">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c14">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="partit_2.html#E36:15_1_3"><i><font color="Green" title="E36">A32</font></i></a>, <a class="txt" href="partit_2.html#E56:15_1_3"><i><font color="Green" title="E45">A62</font></i></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></i>;<br/><b>hence </b><a NAME="E63:15_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c4">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">*</a> <span class="p1">(<span class="default"><a href="partit1.html#K4" title="PARTIT1:func.4">ERl</a> <span class="p2">(<span class="default"><a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c3">P</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="partit_2.html#E60:15_1_3"><i><font color="Green" title="E47">A66</font></i></a>, <a class="ref" href="relset_1.html#T51" title="RELSET_1:th.51">RELSET_1:51</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
