<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">G</font> being  <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>  st <font color="Olive" title="b1">G</font> is  <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a>  holds <br/>for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font> being  <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>  st <font color="Olive" title="b2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Olive" title="b3">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> holds <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="Olive" title="b2">P</font></span>)</span></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</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="Olive" title="b3">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="Olive" title="b3">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</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="Olive" title="b2">P</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">G</font> is  <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a>  implies for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <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>  st <font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> holds <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="Olive" title="b1">P</font></span>)</span></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</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="Olive" title="b2">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="Olive" title="b2">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</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="Olive" title="b1">P</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c2">G</font> is  <a href="bvfunc_2.html#V2" title="BVFUNC_2:attr.2">independent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <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>  st <font color="Olive" title="b1">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Olive" title="b2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> holds <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="Olive" title="b1">P</font></span>)</span></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</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="Olive" title="b2">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="Olive" title="b2">Q</font></span>)</span></span>)</span> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</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="Olive" title="b1">P</font></span>)</span></span>)</span></span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font> &amp; <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> implies <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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:17"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:17"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_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="17_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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><br/><div class="add"><a NAME="E2:17_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<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/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></span>
<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#K4" title="RELSET_1:func.4">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></span>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></span>
;<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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></span>
<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#K4" title="RELSET_1:func.4">*</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="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></span>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:17_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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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><br/><div class="add"><a NAME="E2:17_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<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/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></span>
<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#K4" title="RELSET_1:func.4">*</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></span>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></span>
;<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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="partit_2.html#T1" target="_self" title="PARTIT_2:th.1">Th1</a></span>
<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#K4" title="RELSET_1:func.4">*</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="kw">by</span> <span class="lab"><a class="ref" href="partit1.html#T37" title="PARTIT1:th.37">PARTIT1:37</a></span>
<br/>.= 
 <a href="eqrel_1.html#K1" title="EQREL_1:func.1">nabla</a> <font color="Maroon" title="c1">Y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T4" target="_self" title="PARTIT_2:th.4">Th4</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:17_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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3"><span class="kw">suppose </span></a><a NAME="E1:17_1_3"/><span class="lab"><font color="Green" title="E14">A8</font></span>: 
( <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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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#K4" title="RELSET_1:func.4">*</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#K4" title="RELSET_1:func.4">*</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><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">G9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relset_1.html#D2" title="RELSET_1:def.2">RELSET_1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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#K4" title="RELSET_1:func.4">*</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> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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#K4" title="RELSET_1:func.4">*</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><br/><span class="kw">assume </span><a NAME="E3:17_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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#K4" title="RELSET_1:func.4">*</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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#K4" title="RELSET_1:func.4">*</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><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E5:17_1_3"/><span class="lab"><font color="Green" title="E15">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">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>
 <span class="kw">and </span><br/><a NAME="E6:17_1_3"/><span class="lab"><font color="Green" title="E16">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c7">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T51" title="RELSET_1:th.51">RELSET_1:51</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">A</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E8:17_1_3"/><span class="lab"><font color="Green" title="E17">A11</font></span>: 
<font color="Maroon" title="c9">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>
 <span class="kw">and </span><br/><a NAME="E9:17_1_3"/><span class="lab"><font color="Green" title="E18">A12</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">and </span><br/><a NAME="E10:17_1_3"/><span class="lab"><font color="Green" title="E19">A13</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_3"><span class="lab"><font color="Green" title="E15">A9</font></span></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">B</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Y</font><span class="kw"> such that </span><br/><a NAME="E12:17_1_3"/><span class="lab"><font color="Green" title="E20">A14</font></span>: 
<font color="Maroon" title="c10">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>
 <span class="kw">and </span><br/><a NAME="E13:17_1_3"/><span class="lab"><font color="Green" title="E21">A15</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">and </span><br/><a NAME="E14:17_1_3"/><span class="lab"><font color="Green" title="E22">A16</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_1_3"><span class="lab"><font color="Green" title="E16">A10</font></span></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">hQ</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c12">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><span class="kw"> such that </span><br/><a NAME="E16:17_1_3"/><span class="lab"><font color="Green" title="E23">A17</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">hQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">Q</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">hQ</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">FQ</font> )
 <span class="kw">and </span><br/><a NAME="E17:17_1_3"/><span class="lab"><font color="Green" title="E24">A18</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Q</font> holds <br/><font color="Maroon" title="c11">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>
 <span class="kw">and </span><br/><a NAME="E18:17_1_3"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
<font color="Maroon" title="c10">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="c12">FQ</font>
 <span class="kw">and </span><br/><a NAME="E19:17_1_3"/>
<font color="Maroon" title="c10">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:17_1_3"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">hP</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c14">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><span class="kw"> such that </span><br/><a NAME="E21:17_1_3"/><span class="lab"><font color="Green" title="E26">A20</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">hP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">hP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">FP</font> )
 <span class="kw">and </span><br/><a NAME="E22:17_1_3"/><span class="lab"><font color="Green" title="E27">A21</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/><font color="Maroon" title="c13">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>
 <span class="kw">and </span><br/><a NAME="E23:17_1_3"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
<font color="Maroon" title="c9">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="c14">FP</font>
 <span class="kw">and </span><br/><a NAME="E24:17_1_3"/>
<font color="Maroon" title="c9">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_1_3"><span class="lab"><font color="Green" title="E17">A11</font></span></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">P</font> = <font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c16">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3"><span class="lab"><font color="Green" title="E14">A8</font></span></a></span>;<br/><span class="kw">deffunc </span><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="c15">P</font>) <span class="kw">-&gt; </span>   <a href="bvfunc_1.html#M1" title="BVFUNC_1:mode.1">Element</a> of $1 =  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c7">y</font>,$1;<br/><span class="kw">consider </span><font color="Maroon" title="c17">hP9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c15">P</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E27:17_1_3"/><span class="lab"><font color="Green" title="E29">A23</font></span>: 
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">P</font> holds  <font color="Maroon" title="c17">hP9</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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span><span class="kw">deffunc </span><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="c16">Q</font>) <span class="kw">-&gt; </span>   <a href="bvfunc_1.html#M1" title="BVFUNC_1:mode.1">Element</a> of $1 =  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,$1;<br/><span class="kw">consider </span><font color="Maroon" title="c18">hQ9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c16">Q</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E29:17_1_3"/><span class="lab"><font color="Green" title="E30">A24</font></span>: 
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="c16">Q</font> holds  <font color="Maroon" title="c18">hQ9</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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span><span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/><a NAME="E30:17_1_3"/><span class="lab"><font color="Green" title="E31">A25</font></span>: 
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="c5">G9</font>  holds  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="17_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">d</font> be    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c5">G9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="Maroon" title="c19">d</font>)</span><br/>

<a NAME="E1:17_1_3_1"/>
<font color="Maroon" title="c19">d</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <font color="Maroon" title="c19">d</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E2:17_1_3_1"/>
 <a href="setfam_1.html#K9" title="SETFAM_1:func.9">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="c19">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">consider </span><font color="Maroon" title="c19">h9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c5">G9</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E32:17_1_3"/><span class="lab"><font color="Green" title="E32">A26</font></span>: 
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="c5">G9</font> holds  <font color="Maroon" title="c19">h9</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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S11" title="FUNCT_2:sch.11">FUNCT_2:sch 11</a>(<a class="txt" href="#E30:17_1_3"><span class="lab"><font color="Green" title="E31">A25</font></span></a>);<br/></span><span class="kw">set </span><font color="Maroon" title="c20">h</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font>;<br/><a NAME="E33:17_1_3"/><span class="lab"><font color="Green" title="E33">A27</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">hQ9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E34:17_1_3"/><span class="lab"><font color="Green" title="E34">A28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">hP9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E35:17_1_3"/><span class="lab"><font color="Green" title="E35">A29</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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="c17">hP9</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="17_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c21">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3_2"/><span class="lab"><font color="Green" title="E36">A30</font></span>: 
<font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c22">d9</font> = <font color="Maroon" title="c21">d</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c15">P</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_1_3_2_1"/>
( <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font> or  not <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_2_1_1"/><span class="lab"><font color="Green" title="E37">A31</font></span>: 
<font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font></span><br/><div class="add"><a NAME="E2:17_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A32</font></span>: 
<font color="Maroon" title="c11">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">FQ</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:17_1_3"><span class="lab"><font color="Green" title="E23">A17</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E3:17_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A33</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:17_1_3"><span class="lab"><font color="Green" title="E22">A16</font></span></a>, <a class="txt" href="#E18:17_1_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E4:17_1_3_2_1_1"/><span class="lab"><font color="Green" title="E40">A34</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:17_1_3"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="txt" href="#E18:17_1_3"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E2:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E38">A32</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E5:17_1_3_2_1_1"/><span class="lab"><font color="Green" title="E41">A35</font></span>: 
<font color="Maroon" title="c11">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:17_1_3"><span class="lab"><font color="Green" title="E24">A18</font></span></a>, <a class="txt" href="#E1:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E37">A31</font></span></a></span>;<br/><a NAME="E6:17_1_3_2_1_1"/><span class="lab"><font color="Green" title="E42">A36</font></span>: 
<font color="Maroon" title="c13">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">FP</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:17_1_3"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E1:17_1_3_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E7:17_1_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A37</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_3"><span class="lab"><font color="Green" title="E18">A12</font></span></a>, <a class="txt" href="#E23:17_1_3"><span class="lab"><font color="Green" title="E28">A22</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E8:17_1_3_2_1_1"/><span class="lab"><font color="Green" title="E44">A38</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:17_1_3"><span class="lab"><font color="Green" title="E19">A13</font></span></a>, <a class="txt" href="#E23:17_1_3"><span class="lab"><font color="Green" title="E28">A22</font></span></a>, <a class="txt" href="#E6:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E42">A36</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E9:17_1_3_2_1_1"/><span class="lab"><font color="Green" title="E45">A39</font></span>: 
<font color="Maroon" title="c13">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:17_1_3"><span class="lab"><font color="Green" title="E27">A21</font></span></a>, <a class="txt" href="#E1:17_1_3_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> = 
<font color="Maroon" title="c18">hQ9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17_1_3"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E1:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E37">A31</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c22">d9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:17_1_3"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="txt" href="#E1:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E37">A31</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c13">hP</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E7:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>
<br/>.= 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c22">d9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E8:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c11">hQ</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E4:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E40">A34</font></span></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>
<br/>.= 
 <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c22">d9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E3:17_1_3_2_1_1"><span class="lab"><font color="Green" title="E39">A33</font></span></a>, <a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c17">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17_1_3"><span class="lab"><font color="Green" title="E29">A23</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_2_1_2"/>
not <font color="Maroon" title="c21">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font></span><br/><div class="add"><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17_1_3"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<font color="Maroon" title="c17">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:17_1_3"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E1:17_1_3_2"><span class="lab"><font color="Green" title="E36">A30</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c21">FP9</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">hP9</font>, <font color="Maroon" title="c22">FQ9</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">hQ9</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/><span class="kw">set </span><font color="Maroon" title="c23">A9</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c21">FP9</font>;<br/><span class="kw">set </span><font color="Maroon" title="c24">B9</font> =  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c22">FQ9</font>;<br/><a NAME="E37:17_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="c21">FP9</font> holds <br/><font color="Maroon" title="c7">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="17_1_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">FP9</font> implies <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3_3"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">FP9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:17_1_3_3"/><span class="lab"><font color="Green" title="E36">A40</font></span>: 
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">hP9</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_3_3"/><span class="lab"><font color="Green" title="E37">A41</font></span>: 
<font color="Maroon" title="c17">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">b</font> = <font color="Maroon" title="c26">b</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c15">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_3_3"><span class="lab"><font color="Green" title="E36">A40</font></span></a></span>;<br/>
<a NAME="E6:17_1_3_3"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c27">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17_1_3"><span class="lab"><font color="Green" title="E29">A23</font></span></a>, <a class="txt" href="#E4:17_1_3_3"><span class="lab"><font color="Green" title="E37">A41</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:17_1_3_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E38:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A42</font></span>: 
<font color="Maroon" title="c7">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="c21">FP9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E39:17_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="c22">FQ9</font> holds <br/><font color="Maroon" title="c6">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="17_1_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FQ9</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3_4"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FQ9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c26">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:17_1_3_4"/><span class="lab"><font color="Green" title="E37">A43</font></span>: 
<font color="Maroon" title="c26">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">hQ9</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_3_4"/><span class="lab"><font color="Green" title="E38">A44</font></span>: 
<font color="Maroon" title="c18">hQ9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">b</font> = <font color="Maroon" title="c26">b</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c16">Q</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_3_4"><span class="lab"><font color="Green" title="E37">A43</font></span></a></span>;<br/>
<a NAME="E6:17_1_3_4"/>
<font color="Maroon" title="c25">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c27">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:17_1_3"><span class="lab"><font color="Green" title="E30">A24</font></span></a>, <a class="txt" href="#E4:17_1_3_4"><span class="lab"><font color="Green" title="E38">A44</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:17_1_3_4"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D8" title="EQREL_1:def.8">EQREL_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E40:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A45</font></span>: 
<font color="Maroon" title="c6">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="c22">FQ9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E41:17_1_3"/><span class="lab"><font color="Green" title="E38">A46</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font> holds <br/><font color="Maroon" title="c18">hQ9</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="17_1_3_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c25">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font> implies <font color="Maroon" title="c18">hQ9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">d</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3_5"/>
<font color="Maroon" title="c25">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">hQ9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">d</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">d</font> = <font color="Maroon" title="c25">d</font> as    <a href="partit_2.html#M1" title="PARTIT_2:mode.1">Element</a> of <font color="Maroon" title="c16">Q</font> ;<br/>
<a NAME="E3:17_1_3_5"/>
<font color="Maroon" title="c18">hQ9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c26">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:17_1_3"><span class="lab"><font color="Green" title="E30">A24</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17_1_3_5"/>
<font color="Maroon" title="c18">hQ9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E42:17_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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">h9</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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">hP9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/><a NAME="E43:17_1_3"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E44:17_1_3"/><span class="kw">then </span>
(  <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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> &amp; <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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</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#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c1">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><span class="lab"><font color="Green" title="E39">A47</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17_1_3"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">h9</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">P</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">Q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:17_1_3"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<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="c15">P</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c2">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
;<br/><a NAME="E47:17_1_3"/><span class="lab"><font color="Green" title="E40">A48</font></span>: 
for <font color="Olive" title="b1">d</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font> holds <br/><font color="Maroon" title="c17">hP9</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="17_1_3_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font> implies <font color="Maroon" title="c17">hP9</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3_7"/>
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">hP9</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></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><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="c15">P</font> ;<br/>
<a NAME="E3:17_1_3_7"/>
<font color="Maroon" title="c17">hP9</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bvfunc_1.html#K19" title="BVFUNC_1:func.19">EqClass</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c27">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:17_1_3"><span class="lab"><font color="Green" title="E29">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:17_1_3_7"/>
<font color="Maroon" title="c17">hP9</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E48:17_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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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="17_1_3_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_1_3_8"/><span class="lab"><font color="Green" title="E41">A49</font></span>: 
<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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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></span><br/>

<font color="Maroon" title="c2">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>
<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="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">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="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
;<br/>
<a NAME="E3:17_1_3_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A50</font></span>: 
( <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="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">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="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3_8"><span class="lab"><font color="Green" title="E41">A49</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_8_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:17_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="c16">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="c15">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="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_3_8"><span class="lab"><font color="Green" title="E42">A50</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_8_2_1"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_8_2_1"/><span class="lab"><font color="Green" title="E43">A51</font></span>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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></span><br/><div class="add"><a NAME="E2:17_1_3_8_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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="c18">hQ9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:17_1_3"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_1_3_8_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:17_1_3"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E1:17_1_3_8_2_1"><span class="lab"><font color="Green" title="E43">A51</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_8_2_2"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_8_2_2"/><span class="lab"><font color="Green" title="E43">A52</font></span>: 
<font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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></span><br/><div class="add"><a NAME="E2:17_1_3_8_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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">hP9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:17_1_3"><span class="lab"><font color="Green" title="E35">A29</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_1_3_8_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:17_1_3"><span class="lab"><font color="Green" title="E40">A48</font></span></a>, <a class="txt" href="#E1:17_1_3_8_2_2"><span class="lab"><font color="Green" title="E43">A52</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_3_8_2_3"><span class="kw">suppose </span></a><a NAME="E1:17_1_3_8_2_3"/><span class="lab"><font color="Green" title="E43">A53</font></span>: 
<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="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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></span><br/><div class="add"><a NAME="E2:17_1_3_8_2_3"/><span class="kw">then </span>
not <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c16">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:17_1_3_8_2_3"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hP9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> &amp; not <font color="Maroon" title="c26">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hP9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:17_1_3"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E33:17_1_3"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="funct_4.html#T15" title="FUNCT_4:th.15">FUNCT_4:15</a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>;<br/><a NAME="E4:17_1_3_8_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A54</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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="c19">h9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>;<br/><a NAME="E5:17_1_3_8_2_3"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_3_8_2_3"><span class="lab"><font color="Green" title="E43">A53</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:17_1_3_8_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:17_1_3"><span class="lab"><font color="Green" title="E32">A26</font></span></a>, <a class="txt" href="#E4:17_1_3_8_2_3"><span class="lab"><font color="Green" title="E44">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E49:17_1_3"/><span class="kw">then </span>
 <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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E46:17_1_3"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="ref" href="bvfunc_2.html#D5" title="BVFUNC_2:def.5">BVFUNC_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E51:17_1_3"/><span class="lab"><font color="Green" title="E41">A55</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></span>;<br/><a NAME="E52:17_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="c21">FP9</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="17_1_3_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">FP9</font> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_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="c21">FP9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:17_1_3_9"/><span class="lab"><font color="Green" title="E42">A56</font></span>: 
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c17">hP9</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_3_9"/><span class="lab"><font color="Green" title="E43">A57</font></span>: 
<font color="Maroon" title="c17">hP9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:17_1_3_9"/>
<font color="Maroon" title="c17">hP9</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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:17_1_3"><span class="lab"><font color="Green" title="E35">A29</font></span></a>, <a class="txt" href="#E3:17_1_3_9"><span class="lab"><font color="Green" title="E42">A56</font></span></a></span>;<br/>
<a NAME="E6:17_1_3_9"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E34:17_1_3"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="txt" href="#E46:17_1_3"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E3:17_1_3_9"><span class="lab"><font color="Green" title="E42">A56</font></span></a>, <a class="txt" href="#E4:17_1_3_9"><span class="lab"><font color="Green" title="E43">A57</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:17_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:17_1_3"><span class="lab"><font color="Green" title="E41">A55</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E53:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A58</font></span>: 
<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="c21">FP9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E54:17_1_3"/><span class="kw">then </span>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c21">FP9</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">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:17_1_3"><span class="lab"><font color="Green" title="E40">A48</font></span></a>, <a class="txt" href="#E34:17_1_3"><span class="lab"><font color="Green" title="E34">A28</font></span></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></span>;<br/><a NAME="E55:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A59</font></span>: 
<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="c7">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="c15">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:17_1_3"><span class="lab"><font color="Green" title="E36">A42</font></span></a>, <a class="txt" href="#E53:17_1_3"><span class="lab"><font color="Green" title="E42">A58</font></span></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></span>;<br/><a NAME="E56:17_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="c22">FQ9</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="17_1_3_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c27">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">FQ9</font> implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_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="c22">FQ9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">a</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c28">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:17_1_3_10"/><span class="lab"><font color="Green" title="E44">A60</font></span>: 
<font color="Maroon" title="c28">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">hQ9</font>
 <span class="kw">and </span><br/><a NAME="E4:17_1_3_10"/><span class="lab"><font color="Green" title="E45">A61</font></span>: 
<font color="Maroon" title="c18">hQ9</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><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="c16">Q</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_3_10"><span class="lab"><font color="Green" title="E44">A60</font></span></a></span>;<br/>
<a NAME="E6:17_1_3_10"/>
<font color="Maroon" title="c18">hQ9</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="c19">h9</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">hP9</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">hQ9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1_3_10"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>;<br/>
<a NAME="E7:17_1_3_10"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E33:17_1_3"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="txt" href="#E46:17_1_3"><span class="lab"><font color="Green" title="E39">A47</font></span></a>, <a class="txt" href="#E3:17_1_3_10"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="txt" href="#E4:17_1_3_10"><span class="lab"><font color="Green" title="E45">A61</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:17_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:17_1_3"><span class="lab"><font color="Green" title="E41">A55</font></span></a>, <a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E57:17_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A62</font></span>: 
<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="c22">FQ9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></span>;<br/><a NAME="E58:17_1_3"/><span class="kw">then </span>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c22">FQ9</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="c16">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:17_1_3"><span class="lab"><font color="Green" title="E38">A46</font></span></a>, <a class="txt" href="#E33:17_1_3"><span class="lab"><font color="Green" title="E33">A27</font></span></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></span>;<br/><a NAME="E59:17_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">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="c16">Q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:17_1_3"><span class="lab"><font color="Green" title="E37">A45</font></span></a>, <a class="txt" href="#E57:17_1_3"><span class="lab"><font color="Green" title="E44">A62</font></span></a>, <a class="ref" href="partit1.html#D6" title="PARTIT1:def.6">PARTIT1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E60:17_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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#K4" title="RELSET_1:func.4">*</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="kw">by</span> <span class="lab"><a class="txt" href="#E55:17_1_3"><span class="lab"><font color="Green" title="E43">A59</font></span></a>, <a class="ref" href="relset_1.html#T51" title="RELSET_1:th.51">RELSET_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
