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

<b>let </b><font color="Maroon" title="c1">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">G</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="bvfunc_2.html#K1" title="BVFUNC_2:func.1">PARTITIONS</a> <font color="Maroon" title="c1">Y</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font>, <font color="Maroon" title="c5">D</font> be    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of <font color="Maroon" title="c1">Y</font>;<br/>









<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E6">A1</font></i>: 
( <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c5">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>

<a NAME="E2:7"/><i><font color="Green" title="E7">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c5">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c5">D</font>
 ;<br/>

<a NAME="E2:7_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="partit1.html#D4" title="PARTIT1:def.4">PARTIT1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:7_1"/><i><font color="Green" title="E9">A4</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c8">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">d</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E5:7_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E4:7_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="partit1.html#D4" title="PARTIT1:def.4">PARTIT1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:7_1"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">c</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">h</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span>;<br/>
<a NAME="E8:7_1"/><i><font color="Green" title="E11">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>,<font color="Maroon" title="c5">D</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E3"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<a NAME="E9:7_1"/><i><font color="Green" title="E12">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">d</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</a></i>;<br/>
<a NAME="E10:7_1"/><i><font color="Green" title="E13">A8</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/>
<a NAME="E11:7_1"/><i><font color="Green" title="E14">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E4"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/>
<a NAME="E12:7_1"/><i><font color="Green" title="E15">A10</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">p</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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">p</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_1"/><i><font color="Green" title="E16">A11</font></i>: 
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_1_1_1"/>
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font> or <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> )
 </span><b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_1"><i><font color="Green" title="E16">A11</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_1"><b>case </b></a><a NAME="E1:7_1_1_1_1_1"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E4:7_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_2"><b>case </b></a><a NAME="E1:7_1_1_1_1_2"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E7:7_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="bvfunc14.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1_1_1_3"><b>case </b></a><a NAME="E1:7_1_1_1_1_3"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_1_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E7:7_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="bvfunc14.html#E4"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">p</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:7_1"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E8:7_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<i><font color="Green" title="E17">A13</font></i>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> = 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="txt" href="bvfunc14.html#E6"><i><font color="Green" title="E5">Lm4</font></i></a></i>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T100" title="ENUMSET1:th.100">ENUMSET1:100</a></i>
;<br/>
<a NAME="E15:7_1"/>
 <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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_3"/><i><font color="Green" title="E18">A14</font></i>: 
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_3_1_1"/>
( <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> or <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> )
 </span><b>by </b><i><a class="txt" href="bvfunc14.html#E14:7_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_3"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_1"><b>case </b></a><a NAME="E1:7_1_3_1_1_1"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_3_1_1_1"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E4:7_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="bvfunc14.html#E9:7_1"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_2"><b>case </b></a><a NAME="E1:7_1_3_1_1_2"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><a NAME="E2:7_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:7_1_3_1_1_2"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E7:7_1"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_3_1_1_3"><b>case </b></a><a NAME="E1:7_1_3_1_1_3"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><a NAME="E2:7_1_3_1_1_3"/><b>then </b>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">c</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E4"><i><font color="Green" title="E3">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E3:7_1_3_1_1_3"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E7:7_1"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_3"/>
<font color="Maroon" title="c12">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Y</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">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"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></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> ;<br/>
<a NAME="E17:7_1"/><i><font color="Green" title="E18">A15</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E13:7_1"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E18:7_1"/><i><font color="Green" title="E19">A16</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c12">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_4"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 ;<br/>

<a NAME="E2:7_1_4"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F</font> holds <br/><font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_1_4_1"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_4_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_4_1_1_1"/>
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> or <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> )
 </span><b>by </b><i><a class="txt" href="bvfunc14.html#E14:7_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_4_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:7_1_4_1_1_1_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_4_1_1_1_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E4:7_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="bvfunc14.html#E9:7_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_4"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:7_1_4_1_1_1_2"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><a NAME="E2:7_1_4_1_1_1_2"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">c</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E4:7_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="bvfunc14.html#E7:7_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_4"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/><b>hence </b><a NAME="E3:7_1_4_1_1_1_2"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E10:7_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_4_1_1_1_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:7_1_4_1_1_1_3"/><i><font color="Green" title="E22">A20</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><a NAME="E2:7_1_4_1_1_1_3"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">d</font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E4:7_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="bvfunc14.html#E7:7_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_4"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/><b>hence </b><a NAME="E3:7_1_4_1_1_1_3"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E11:7_1"><i><font color="Green" title="E14">A9</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_1_4_1_1_1_3"><i><font color="Green" title="E22">A20</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_4_1"/>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:7_1_4"/><b>then </b>
<font color="Maroon" title="c13">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c12">F</font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E14:7_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:7_1_4"/>
<font color="Maroon" title="c13">u</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="c12">F</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E14:7_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:7_1"/>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c12">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_5"/>
<font color="Maroon" title="c13">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="c12">F</font>
 ;<br/>

<a NAME="E2:7_1_5"/><b>then </b><i><font color="Green" title="E20">A21</font></i>: 
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E17:7_1"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E3:7_1_5"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E4:7_1_5"/><b>then </b><i><font color="Green" title="E21">A22</font></i>: 
( <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">C</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c10">c</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">D</font> <a href="funcop_1.html#K14" title="FUNCOP_1:func.14">.--&gt;</a> <font color="Maroon" title="c8">d</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E14:7_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="bvfunc14.html#E2:7_1_5"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E5:7_1_5"/><b>then </b>
( <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">c</font> &amp; <font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">d</font> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E4"><i><font color="Green" title="E3">Lm2</font></i></a>, <a class="txt" href="bvfunc14.html#E5"><i><font color="Green" title="E4">Lm3</font></i></a>, <a class="ref" href="funct_7.html#T96" title="FUNCT_7:th.96">FUNCT_7:96</a></i>;<br/>
<a NAME="E6:7_1_5"/><b>then </b>
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">b</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">c</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:7_1_5"/>
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E4:7_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="txt" href="bvfunc14.html#E7:7_1"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="bvfunc14.html#E9:7_1"><i><font color="Green" title="E12">A7</font></i></a>, <a class="txt" href="bvfunc14.html#E4:7_1_5"><i><font color="Green" title="E21">A22</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:7_1"/><b>then </b><i><font color="Green" title="E20">A23</font></i>: 
<font color="Maroon" title="c6">x</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">F</font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E18:7_1"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E21:7_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E1:7_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E22:7_1"/>
<font color="Maroon" title="c6">x</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="c2">G</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E8:7_1"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="bvfunc14.html#E12:7_1"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="bvfunc14.html#E20:7_1"><i><font color="Green" title="E20">A23</font></i></a>, <a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:7"/>
 <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c2">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c5">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_2"/>
<font color="Maroon" title="c6">x</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="c2">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c8">F</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:7_2"/><i><font color="Green" title="E8">A24</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F</font> &amp; ( 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/><font color="Maroon" title="c7">h</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> ) &amp; <font color="Maroon" title="c6">x</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="c8">F</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="bvfunc_2.html#D1" title="BVFUNC_2:def.1">BVFUNC_2:def 1</a></i>;<br/>
<a NAME="E4:7_2"/>
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E5:7_2"/><b>then </b><i><font color="Green" title="E9">A25</font></i>: 
( <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D</font> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2"><i><font color="Green" title="E8">A24</font></i></a></i>;<br/>
<a NAME="E6:7_2"/>
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">h</font> &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">h</font> &amp; <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">h</font> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E3:7_2"><i><font color="Green" title="E8">A24</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E7:7_2"/><b>then </b><i><font color="Green" title="E10">A26</font></i>: 
( <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> &amp; <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> &amp; <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:7_2"/><i><font color="Green" title="E11">A27</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2"><i><font color="Green" title="E8">A24</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:7_2"/><i><font color="Green" title="E12">A28</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_2_1"/><i><font color="Green" title="E13">A29</font></i>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span>
 ;<br/>

<a NAME="E2:7_2_1"/><b>then </b><i><font color="Green" title="E14">A30</font></i>: 
( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span> &amp; <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:7_2_1"/><i><font color="Green" title="E15">A31</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_2_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:7_2_1_1"/><i><font color="Green" title="E16">A32</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">h</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_2_1_1_1_1"/>
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font> )
 </span><b>by </b><i><a class="txt" href="bvfunc14.html#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="bvfunc14.html#E3:7_2"><i><font color="Green" title="E8">A24</font></i></a>, <a class="txt" href="bvfunc14.html#E3:7_2_1_1"><i><font color="Green" title="E16">A32</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:7_2_1_1_1_1_1"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_1_1_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2_1_1"><i><font color="Green" title="E16">A32</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:7_2_1_1_1_1_2"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_1_1_1_2"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2_1_1"><i><font color="Green" title="E16">A32</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_1_1_1_3"><b>case </b></a><a NAME="E1:7_2_1_1_1_1_3"/>
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_1_1_1_3"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2_1_1"><i><font color="Green" title="E16">A32</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:7_2_1_1"/>
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:7_2_1"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> holds <br/><font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7_2_1_2"/><i><font color="Green" title="E16">A33</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_2_1_2_1_1"/>
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> or <font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> )
 </span><b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2_1"><i><font color="Green" title="E15">A31</font></i></a>, <a class="txt" href="bvfunc14.html#E1:7_2_1_2"><i><font color="Green" title="E16">A33</font></i></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_1_1_1"><b>case </b></a><a NAME="E1:7_2_1_2_1_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_2_1_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E2:7_2_1"><i><font color="Green" title="E14">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_1_1_2"><b>case </b></a><a NAME="E1:7_2_1_2_1_1_2"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_2_1_1_2"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E2:7_2_1"><i><font color="Green" title="E14">A30</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_2_1_2_1_1_3"><b>case </b></a><a NAME="E1:7_2_1_2_1_1_3"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_2_1_2_1_1_3"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E1:7_2_1"><i><font color="Green" title="E13">A29</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_2_1_2"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">y</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:7_2_1"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E7:7_2"><i><font color="Green" title="E10">A26</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:7_2_1"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2"><i><font color="Green" title="E8">A24</font></i></a>, <a class="txt" href="bvfunc14.html#E7:7_2"><i><font color="Green" title="E10">A26</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:7_2"/><i><font color="Green" title="E13">A34</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_2_2"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">x</font>
 ;<br/>

<a NAME="E2:7_2_2"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2"><i><font color="Green" title="E8">A24</font></i></a>, <a class="txt" href="bvfunc14.html#E7:7_2"><i><font color="Green" title="E10">A26</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E3:7_2_2"/><b>then </b><i><font color="Green" title="E14">A35</font></i>: 
( <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> )
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E7:7_2"><i><font color="Green" title="E10">A26</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E4:7_2_2"/><b>then </b>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:7_2_2"/>
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2_2"><i><font color="Green" title="E14">A35</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:7_2"/><b>then </b><i><font color="Green" title="E14">A36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E9:7_2"><i><font color="Green" title="E12">A28</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">m</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span>;<br/>
<a NAME="E12:7_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E3:7_2"><i><font color="Green" title="E8">A24</font></i></a>, <a class="txt" href="bvfunc14.html#E9:7_2"><i><font color="Green" title="E12">A28</font></i></a>, <a class="txt" href="bvfunc14.html#E10:7_2"><i><font color="Green" title="E13">A34</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E13:7_2"/><b>then </b><i><font color="Green" title="E15">A37</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:7_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E5:7_2"><i><font color="Green" title="E9">A25</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E15:7_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E13:7_2"><i><font color="Green" title="E15">A37</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E16:7_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font>
 
<b>by </b><i><a class="ref" href="partit1.html#D4" title="PARTIT1:def.4">PARTIT1:def 4</a></i>;<br/>
<a NAME="E17:7_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span>,<font color="Maroon" title="c5">D</font>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E5:7_2"><i><font color="Green" title="E9">A25</font></i></a>, <a class="txt" href="bvfunc14.html#E11:7_2"><i><font color="Green" title="E14">A36</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E18:7_2"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span>,<font color="Maroon" title="c5">D</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="bvfunc14.html#E8:7_2"><i><font color="Green" title="E11">A27</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E19:7_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="ref" href="partit1.html#D4" title="PARTIT1:def.4">PARTIT1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
 <a href="bvfunc_2.html#K2" title="BVFUNC_2:func.2">'/\'</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="partit1.html#K2" title="PARTIT1:func.2">'/\'</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="bvfunc14.html#E2:7"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
