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

<a NAME="E1:27_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> &amp; <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> holds <br/><font color="Olive" title="b1">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> implies <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> )</span><br/>



<b>assume </b><a NAME="E1:27_1_1_1"/><i><font color="Green" title="E21">B0</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font></span><br/>

<b>then consider </b><font color="Maroon" title="c8">E</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c9">f</font> being    <a href="kolmog01.html#M2" title="KOLMOG01:mode.2">SigmaSection</a> of <font color="Maroon" title="c8">E</font>,<font color="Maroon" title="c4">F</font><b> such that </b><br/><a NAME="E3:27_1_1_1"/><i><font color="Green" title="E22">B1</font></i>: 
( <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">J</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="kolmog01.html#D9" target="_self" title="KOLMOG01:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">E'</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c11">f'</font> being    <a href="kolmog01.html#M2" title="KOLMOG01:mode.2">SigmaSection</a> of <font color="Maroon" title="c10">E'</font>,<font color="Maroon" title="c4">F</font><b> such that </b><br/><a NAME="E5:27_1_1_1"/><i><font color="Green" title="E23">B2</font></i>: 
( <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">J</font> &amp; <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1"><i><font color="Green" title="E21">B0</font></i></a>, <a class="ref" href="kolmog01.html#D9" target="_self" title="KOLMOG01:def.9">Def9</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>;<br/>
<b>consider </b><font color="Maroon" title="c12">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E7:27_1_1_1"/><i><font color="Green" title="E24">B3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">i</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">i</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</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="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon" title="c1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c13">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:27_1_1_1"/><i><font color="Green" title="E25">B4</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">i</font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>] implies <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">i</font>) ) ) ) )
 <b>from </b><i><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></i>
<a NAME="E10:27_1_1_1"/><i><font color="Green" title="E26">B5</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font> holds <br/><font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font> implies <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_1"/><i><font color="Green" title="E27">C0</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_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="E1:27_1_1_1_1_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span> or  not <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_1"/><i><font color="Green" title="E28">S</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_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="E1:27_1_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> or  not <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E29">F0</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="txt" href="#E7:27_1_1_1"><i><font color="Green" title="E24">B3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:27_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E29">F0</font></i></a>, <a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E29">F1</font></i>: 
not <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E30">G0</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:27_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E29">F1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="txt" href="#E7:27_1_1_1"><i><font color="Green" title="E24">B3</font></i></a></i>;<br/><b>hence </b><a NAME="E4:27_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E2:27_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E30">G0</font></i></a>, <a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:27_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_1_1_1"><i><font color="Green" title="E28">S</font></i></a>, <a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_1_1_2"/><i><font color="Green" title="E28">D1</font></i>: 
not <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font></span><br/><div class="add"><a NAME="E2:27_1_1_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K4" title="FINSUB_1:func.4">\+\</a> <font color="Maroon" title="c10">E'</font>
 ;<br/><a NAME="E3:27_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">E0</font></i>: 
( <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> iff <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E'</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T1" title="XBOOLE_0:th.1">XBOOLE_0:1</a></i>;<br/><a NAME="E4:27_1_1_1_1_1_2"/><i><font color="Green" title="E30">E2</font></i>: 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_1_1_2"><i><font color="Green" title="E29">E0</font></i></a>, <a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a>, <a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:27_1_1_1_1_1_2"/><i><font color="Green" title="E31">E3</font></i>: 
<font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_1_1_2"><i><font color="Green" title="E29">E0</font></i></a>, <a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a>, <a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">Fi</font> = <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font> as   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">Omega</font> <b>by </b><i><a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="ref" href="kolmog01.html#D2" target="_self" title="KOLMOG01:def.2">Def2</a></i>;<br/><a NAME="E7:27_1_1_1_1_1_2"/><i><font color="Green" title="E32">E4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</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="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c15">Fi</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T49" title="PROB_1:th.49">PROB_1:49</a>, <a class="txt" href="#E4:27_1_1_1_1_1_2"><i><font color="Green" title="E30">E2</font></i></a>, <a class="txt" href="#E5:27_1_1_1_1_1_2"><i><font color="Green" title="E31">E3</font></i></a></i>;<br/><a NAME="E8:27_1_1_1_1_1_2"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</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="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_1_1_2"><i><font color="Green" title="E28">D1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_1"><i><font color="Green" title="E27">C0</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>;<br/><b>hence </b><a NAME="E9:27_1_1_1_1_1_2"/>
<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="#E7:27_1_1_1_1_1_2"><i><font color="Green" title="E32">E4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E11:27_1_1_1"/>
<font color="Maroon" title="c13">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>,<font color="Maroon" title="c3">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Sigma</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2_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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c14">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="c13">g</font> or <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_2_1"/>
<font color="Maroon" title="c14">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="c13">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font></span><br/>

<b>then consider </b><font color="Maroon" title="c15">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1_2_1"/><i><font color="Green" title="E27">E0</font></i>: 
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</font> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</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="E4:27_1_1_1_2_1"/><i><font color="Green" title="E28">E1</font></i>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c15">i</font>
 
<b>by </b><i><a class="txt" href="#E3:27_1_1_1_2_1"><i><font color="Green" title="E27">E0</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="txt" href="#E10:27_1_1_1"><i><font color="Green" title="E26">B5</font></i></a></i>;<br/>
<a NAME="E5:27_1_1_1_2_1"/>
<font color="Maroon" title="c4">F</font> <a href="cantor_1.html#K3" title="CANTOR_1:func.3">.</a> <font color="Maroon" title="c15">i</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="c3">Sigma</font>
 
<b>by </b><i><a class="txt" href="#E3:27_1_1_1_2_1"><i><font color="Green" title="E27">E0</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1_1_1_2_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Sigma</font>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_2_1"><i><font color="Green" title="E28">E1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1_1_2"/>
<font color="Maroon" title="c13">g</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>,<font color="Maroon" title="c3">Sigma</font>
 <b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</font> as    <a href="kolmog01.html#M2" title="KOLMOG01:mode.2">SigmaSection</a> of <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>,<font color="Maroon" title="c4">F</font> <b>by </b><i><a class="txt" href="#E10:27_1_1_1"><i><font color="Green" title="E26">B5</font></i></a>, <a class="ref" href="kolmog01.html#D4" target="_self" title="KOLMOG01:def.4">Def4</a></i>;<br/>
<a NAME="E13:27_1_1_1"/><i><font color="Green" title="E27">B6</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">J</font>
 
<b>by </b><i><a class="txt" href="#E3:27_1_1_1"><i><font color="Green" title="E22">B1</font></i></a>, <a class="txt" href="#E5:27_1_1_1"><i><font color="Green" title="E23">B2</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E14:27_1_1_1"/><i><font color="Green" title="E28">B7</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1_1_3"/><i><font color="Green" title="E29">C0</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_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="E1:27_1_1_1_3_1_1"/>
( <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_1_1_1"/><i><font color="Green" title="E30">E0</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><b>then </b><i><font color="Green" title="E31">F0</font></i>:  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> = 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>

<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>
;<br/><a NAME="E3:27_1_1_1_3_1_1_1"/>
(  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> &amp;  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_1_1_1"><i><font color="Green" title="E30">E0</font></i></a></i>;<br/><b>hence </b><a NAME="E4:27_1_1_1_3_1_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_1_1_1"><i><font color="Green" title="E31">F0</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_1_1_2"/>
not <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c15">EnE'</font> = <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> as  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/><a NAME="E3:27_1_1_1_3_1_1_2"/><i><font color="Green" title="E30">F0</font></i>: 
for <font color="Olive" title="b1">B</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c15">EnE'</font> holds   <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">B</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">B</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c15">EnE'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">B</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">B</font></span>)</span></span>)</span></span>)</span></span><br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">I</font></span>)</span></span>)</span></span>)</span>;<br/>
<a NAME="E1:27_1_1_1_3_1_1_2_1"/><i><font color="Green" title="E31">G0</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c15">EnE'</font>]
 
;<br/>
<a NAME="E2:27_1_1_1_3_1_1_2_1"/><i><font color="Green" title="E32">G1</font></i>: 
for <font color="Olive" title="b1">B'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c15">EnE'</font><br/> for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">EnE'</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">B'</font>] &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B'</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">B'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c15">EnE'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">EnE'</font>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font>] &amp; not <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B'</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><b>let </b><font color="Maroon" title="c18">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c15">EnE'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font>] &amp; not <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B'</font> implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>] )</span><br/>



<b>assume </b><a NAME="E1:27_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E33">H0</font></i>: 
(  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/>

<a NAME="E2:27_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E34">H1</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b2">B'</font>, <font color="Olive" title="b3">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <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"><font color="Olive" title="b1">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">B'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">B'</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds   <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"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">B'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c20">B'</font>, <font color="Maroon" title="c21">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">B'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span><br/>



<a NAME="E1:27_1_1_1_3_1_1_2_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK4" title="RELAT_1:NK.4">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">B'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T107" title="RELAT_1:th.107">RELAT_1:107</a></i>;<br/>
<b>hence </b><a NAME="E2:27_1_1_1_3_1_1_2_1_1_1"/>
 <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"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">B'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c21">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T26" title="RELAT_1:th.26">RELAT_1:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:27_1_1_1_3_1_1_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E35">h2</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E5:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E36">H2</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:27_1_1_1_3_1_1_2_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E37">h3</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E8:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E38">H3</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E9:27_1_1_1_3_1_1_2_1_1"/>
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E10:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E39">H4</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a>, <a class="txt" href="#E5:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E36">H2</font></i></a>, <a class="txt" href="#E8:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E38">H3</font></i></a>, <a class="ref" href="zfmisc_1.html#T52" title="ZFMISC_1:th.52">ZFMISC_1:52</a></i>;<br/>
<a NAME="E11:27_1_1_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c18">b</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"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font> )
 
<b>by </b><i><a class="txt" href="#E4:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E35">h2</font></i></a>, <a class="txt" href="#E7:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E37">h3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E12:27_1_1_1_3_1_1_2_1_1"/><b>then </b>
( <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<a NAME="E13:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E40">H5</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="#E10:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E39">H4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E14:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E41">H6</font></i>: 
(  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> &amp;  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> )
 
<b>by </b><i><a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>;<br/>
<a NAME="E15:27_1_1_1_3_1_1_2_1_1"/><i><font color="Green" title="E42">H7</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E16:27_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E17:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E43">H8</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="txt" href="#E15:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E42">H7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E18:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E44">H9</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E19:27_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">E</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</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/>
<a NAME="E20:27_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K4" title="FINSUB_1:func.4">\+\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T102" title="XBOOLE_1:th.102">XBOOLE_1:102</a></i>;<br/>
<a NAME="E21:27_1_1_1_3_1_1_2_1_1"/><b>then </b>
not <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K4" title="FINSUB_1:func.4">\+\</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<a NAME="E22:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E45">H10</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">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="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E18:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E44">H9</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>;<br/>
<a NAME="E23:27_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E24:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E46">h11</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="#E15:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E42">H7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E25:27_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E26:27_1_1_1_3_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T52" title="ZFMISC_1:th.52">ZFMISC_1:52</a></i>;<br/>
<a NAME="E27:27_1_1_1_3_1_1_2_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E28:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E47">H12</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T14" title="FUNCT_1:th.14">FUNCT_1:14</a></i>;<br/>
<a NAME="E29:27_1_1_1_3_1_1_2_1_1"/>
( <font color="Maroon" title="c18">b</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"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> )
 
<b>by </b><i><a class="txt" href="#E24:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E46">h11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E30:27_1_1_1_3_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/>
<a NAME="E31:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E48">H13</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a>, <a class="txt" href="#E28:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E47">H12</font></i></a></i>;<br/>
<a NAME="E32:27_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E49">H14</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_3_1_1_2_1_1_2"/>
( <font color="Maroon" title="c17">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c17">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1_1_2_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c17">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><div class="add"><b>hence </b><a NAME="E2:27_1_1_1_3_1_1_2_1_1_2_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="#E31:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E48">H13</font></i></a>, <a class="txt" href="#E22:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E45">H10</font></i></a>, <a class="txt" href="#E14:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E41">H6</font></i></a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1_1_2_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_1_1_2_1_1_2_2"/><i><font color="Green" title="E50">J1</font></i>: 
not <font color="Maroon" title="c17">B'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c19">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">B'</font>;<br/><a NAME="E3:27_1_1_1_3_1_1_2_1_1_2_2"/><i><font color="Green" title="E51">K0</font></i>: 
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">B'</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_1_1_2_1_1_2_2"><i><font color="Green" title="E50">J1</font></i></a></i>;<br/><a NAME="E4:27_1_1_1_3_1_1_2_1_1_2_2"/>
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E5:27_1_1_1_3_1_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="txt" href="#E17:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E43">H8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:27_1_1_1_3_1_1_2_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c19">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3_1_1_2_1_1_2_2"><i><font color="Green" title="E51">K0</font></i></a>, <a class="ref" href="relat_1.html#T86" title="RELAT_1:th.86">RELAT_1:86</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:27_1_1_1_3_1_1_2_1_1_2_2"/><i><font color="Green" title="E52">K1</font></i>: 
<font color="Maroon" title="c20">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T18" title="RELAT_1:th.18">RELAT_1:18</a></i>;<br/><a NAME="E9:27_1_1_1_3_1_1_2_1_1_2_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:27_1_1_1_3_1_1_2_1_1_2_2"><i><font color="Green" title="E52">K1</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E28:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E47">H12</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E53">K2</font></i>:  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></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="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E33">H0</font></i></a>, <a class="txt" href="#E32:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E49">H14</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E34">H1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a>, <a class="txt" href="#E22:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E45">H10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
;<br/><a NAME="E11:27_1_1_1_3_1_1_2_1_1_2_2"/><i><font color="Green" title="E54">K3</font></i>: 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1_1_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1_1_3_1_1_2_1_1_2_2_2"/><i><font color="Green" title="E55">L1</font></i>: 
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E2:27_1_1_1_3_1_1_2_1_1_2_2_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:27_1_1_1_3_1_1_2_1_1_2_2_2"/><b>then </b>
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_1_1_2_1_1_2_2_2"><i><font color="Green" title="E55">L1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:27_1_1_1_3_1_1_2_1_1_2_2_2"/><b>then </b><i><font color="Green" title="E56">l2</font></i>: 
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> = 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E34">H1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E39">H4</font></i></a>, <a class="txt" href="#E4:27_1_1_1_3_1_1_2_1_1_2_2_2"><i><font color="Green" title="E56">l2</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_1_1_2_1_1_2_2"><i><font color="Green" title="E50">J1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:27_1_1_1_3_1_1_2_1_1_2_2_2"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></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="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E40">H5</font></i></a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E12:27_1_1_1_3_1_1_2_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></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="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1_1_2_1_1_2_2_3"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1_1_3_1_1_2_1_1_2_2_3"/><i><font color="Green" title="E55">L1</font></i>: 
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E2:27_1_1_1_3_1_1_2_1_1_2_2_3"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:27_1_1_1_3_1_1_2_1_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_1_1_2_1_1_2_2_3"><i><font color="Green" title="E55">L1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:27_1_1_1_3_1_1_2_1_1_2_2_3"/><b>then </b><i><font color="Green" title="E56">l2</font></i>: 
<font color="Maroon" title="c17">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> = 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E34">H1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E10:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E39">H4</font></i></a>, <a class="txt" href="#E4:27_1_1_1_3_1_1_2_1_1_2_2_3"><i><font color="Green" title="E56">l2</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_1_1_2_1_1_2_2"><i><font color="Green" title="E50">J1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:27_1_1_1_3_1_1_2_1_1_2_2_3"/>
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c17">B'</font></span>)</span></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="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:27_1_1_1_3_1_1_2_1_1"><i><font color="Green" title="E40">H5</font></i></a>, <a class="ref" href="setfam_1.html#T11" title="SETFAM_1:th.11">SETFAM_1:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:27_1_1_1_3_1_1_2_1_1_2_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c17">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>]
 <b>by </b><i><a class="txt" href="#E10:27_1_1_1_3_1_1_2_1_1_2_2"><i><font color="Green" title="E53">K2</font></i></a>, <a class="txt" href="#E11:27_1_1_1_3_1_1_2_1_1_2_2"><i><font color="Green" title="E54">K3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:27_1_1_1_3_1_1_2_1"/>
for <font color="Olive" title="b1">B1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c15">EnE'</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">B1</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S2" title="SETWISEO:sch.2">SETWISEO:sch 2</a>(<a class="txt" href="#E1:27_1_1_1_3_1_1_2_1"><i><font color="Green" title="E31">G0</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3_1_1_2_1"><i><font color="Green" title="E32">G1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:27_1_1_1_3_1_1_2_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">B</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">B</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">B</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c16">EE'</font> = <font color="Maroon" title="c15">EnE'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c15">EnE'</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/><a NAME="E5:27_1_1_1_3_1_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">EE'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">EE'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c16">EE'</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_1_1_2"><i><font color="Green" title="E30">F0</font></i></a></i>;<br/><b>hence </b><a NAME="E6:27_1_1_1_3_1_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:27_1_1_1_3"/><i><font color="Green" title="E30">C1</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E31">D0</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">E</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>
<br/>.= 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E32">D1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">E</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font>

<br/>.= 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E3:27_1_1_1_3_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_3_2_3"/><i><font color="Green" title="E33">E0</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/>

<a NAME="E2:27_1_1_1_3_2_3"/><b>then </b><i><font color="Green" title="E34">E2</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_2"><i><font color="Green" title="E31">D0</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:27_1_1_1_3_2_3"/><i><font color="Green" title="E35">e3</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_2_3"><i><font color="Green" title="E34">E2</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_2_3"><i><font color="Green" title="E33">E0</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_2_3"><i><font color="Green" title="E35">e3</font></i></a>, <a class="txt" href="#E7:27_1_1_1"><i><font color="Green" title="E24">B3</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_2_3"><i><font color="Green" title="E33">E0</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_2"><i><font color="Green" title="E31">D0</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:27_1_1_1_3_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_2_3"><i><font color="Green" title="E33">E0</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:27_1_1_1_3_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_2"><i><font color="Green" title="E32">D1</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:27_1_1_1_3"/><i><font color="Green" title="E31">C2</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_3"><b>proof </b></a><div class="add">

<i><font color="Green" title="E32">D0</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>
<br/>.= 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E33">D1</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font>

<br/>.= 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E3:27_1_1_1_3_3"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_3_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_3_3_3"/><i><font color="Green" title="E34">E0</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font></span><br/>

<a NAME="E2:27_1_1_1_3_3_3"/><b>then </b><i><font color="Green" title="E35">E2</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</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>, <a class="txt" href="#E1:27_1_1_1_3_3"><i><font color="Green" title="E32">D0</font></i></a></i>;<br/>
<a NAME="E3:27_1_1_1_3_3_3"/>
not <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_3_3"><i><font color="Green" title="E34">E0</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_3"><i><font color="Green" title="E32">D0</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>then </b><font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> = 
<font color="Maroon" title="c12">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_3_3"><i><font color="Green" title="E34">E0</font></i></a>, <a class="txt" href="#E7:27_1_1_1"><i><font color="Green" title="E24">B3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_3_3"><i><font color="Green" title="E34">E0</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3_3_3"><i><font color="Green" title="E35">E2</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:27_1_1_1_3_3_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_3_3"><i><font color="Green" title="E34">E0</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:27_1_1_1_3_3"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_3"><i><font color="Green" title="E33">D1</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:27_1_1_1_3"/><i><font color="Green" title="E32">C3</font></i>: 
for <font color="Olive" title="b1">E</font>, <font color="Olive" title="b2">E'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">E'</font> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b2">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b2">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">E</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">E</font>, <font color="Maroon" title="c16">E'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font> holds <br/> <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c17">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:27_1_1_1_3_4"/><i><font color="Green" title="E33">D0</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></span><br/>

<b>thus </b><a NAME="E2:27_1_1_1_3_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">y</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c18">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="c17">g</font> or <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_3_4_1"/>
<font color="Maroon" title="c18">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="c17">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c19">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1_3_4_1"/><i><font color="Green" title="E34">F0</font></i>: 
( <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">g</font> &amp; <font color="Maroon" title="c18">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</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="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_3_4_1_1"/>
( <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span> or  not <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_4_1_1_1"/><i><font color="Green" title="E35">G0</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_4_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_4_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_4_1_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="E1:27_1_1_1_3_4_1_1_1_1_1"/>
( <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> or  not <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_4_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_4_1_1_1_1_1_1"/><i><font color="Green" title="E36">J0</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_4_1_1_1_1_1_1"/><i><font color="Green" title="E37">K0</font></i>: 
<font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T49" title="XBOOLE_1:th.49">XBOOLE_1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_4_1_1_1_1_1_1"><i><font color="Green" title="E37">K0</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
;<br/><a NAME="E4:27_1_1_1_3_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_4_1_1_1_1_1_1"><i><font color="Green" title="E36">J0</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_4_1_1_1"><i><font color="Green" title="E35">G0</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:27_1_1_1_3_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_4_1"><i><font color="Green" title="E34">F0</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="E6:27_1_1_1_3_4_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E38">K1</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E7:27_1_1_1_3_4_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_4_1"><i><font color="Green" title="E34">F0</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E8:27_1_1_1_3_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:27_1_1_1_3_4_1_1_1_1_1_1"><i><font color="Green" title="E38">K1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:27_1_1_1_3_4_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></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="E10:27_1_1_1_3_4_1_1_1_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_4_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_4_1_1_1_1_1_2"/><i><font color="Green" title="E36">J1</font></i>: 
not <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">E</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">E</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T42" title="XBOOLE_1:th.42">XBOOLE_1:42</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c15">E</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
<br/>.= 
<font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T41" title="XBOOLE_1:th.41">XBOOLE_1:41</a></i>
;<br/><a NAME="E3:27_1_1_1_3_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c15">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_4_1_1_1_1_1_2"><i><font color="Green" title="E36">J1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_4_1_1_1"><i><font color="Green" title="E35">G0</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:27_1_1_1_3_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_4_1"><i><font color="Green" title="E34">F0</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:27_1_1_1_3_4_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E37">K0</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E6:27_1_1_1_3_4_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_4_1"><i><font color="Green" title="E34">F0</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E7:27_1_1_1_3_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:27_1_1_1_3_4_1_1_1_1_1_2"><i><font color="Green" title="E37">K0</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:27_1_1_1_3_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></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/><a NAME="E9:27_1_1_1_3_4_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></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="E10:27_1_1_1_3_4_1_1_1_1_1_2"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:27_1_1_1_3_4_1_1_1"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_4_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_4_1_1_2"/>
not <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_4_1_1_2"/><b>then </b><i><font color="Green" title="E35">H0</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c16">E'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_4_1"><i><font color="Green" title="E34">F0</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_4"><i><font color="Green" title="E33">D0</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <font color="Maroon" title="c16">E'</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">E</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T102" title="XBOOLE_1:th.102">XBOOLE_1:102</a></i>
<br/>.= 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c15">E</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c16">E'</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="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
;<br/><a NAME="E4:27_1_1_1_3_4_1_1_2"/><b>then </b><i><font color="Green" title="E36">H1</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_4"><i><font color="Green" title="E33">D0</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3_4_1_1_2"><i><font color="Green" title="E35">H0</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/><a NAME="E5:27_1_1_1_3_4_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">y</font>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_4_1"><i><font color="Green" title="E34">F0</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><a NAME="E6:27_1_1_1_3_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">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> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_3_4_1_1_2"><i><font color="Green" title="E36">H1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E7:27_1_1_1_3_4_1_1_2"/><b>then </b>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></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="E8:27_1_1_1_3_4_1_1_2"/>
<font color="Maroon" title="c18">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<a NAME="E3:27_1_1_1_3_4"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<a NAME="E4:27_1_1_1_3_4"/><b>then </b><i><font color="Green" title="E34">E0</font></i>: 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E5:27_1_1_1_3_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99" title="RELAT_1:th.99">RELAT_1:99</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1_1_1_3_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">E</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c16">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">g</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">E'</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c15">E</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">g</font>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_3_4"><i><font color="Green" title="E34">E0</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:27_1_1_1_3"/><i><font color="Green" title="E33">C4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:27_1_1_1_3"><i><font color="Green" title="E32">C3</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_3_5"/>
( <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_1"/><i><font color="Green" title="E34">D0</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_1"/><b>then </b><i><font color="Green" title="E35">e0</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:27_1_1_1_3_5_1"/><i><font color="Green" title="E36">X</font></i>: 
( <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font> &amp; <font color="Maroon" title="c8">E</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <font color="Maroon" title="c10">E'</font> )
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_5_1"><i><font color="Green" title="E34">D0</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E4:27_1_1_1_3_5_1"/><b>then </b><i><font color="Green" title="E37">E1</font></i>: 
( <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T86" title="XBOOLE_1:th.86">XBOOLE_1:86</a></i>;<br/><a NAME="E5:27_1_1_1_3_5_1"/><i><font color="Green" title="E38">E2</font></i>: 
( <font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">f'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T40" title="FUNCT_2:th.40">FUNCT_2:40</a>, <a class="txt" href="#E3:27_1_1_1_3_5_1"><i><font color="Green" title="E36">X</font></i></a>, <a class="ref" href="xboole_1.html#T86" title="XBOOLE_1:th.86">XBOOLE_1:86</a></i>;<br/><a NAME="E6:27_1_1_1_3_5_1"/>
( <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font> &amp; <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E7:27_1_1_1_3_5_1"/><b>then </b><i><font color="Green" title="E39">E3</font></i>: 
( <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> )
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_3_5_1"><i><font color="Green" title="E37">E1</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="E8:27_1_1_1_3_5_1"/>
( <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> &amp; <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> )
 <b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E9:27_1_1_1_3_5_1"/>
<font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1"><i><font color="Green" title="E22">B1</font></i></a>, <a class="txt" href="#E5:27_1_1_1"><i><font color="Green" title="E23">B2</font></i></a>, <a class="txt" href="#E5:27_1_1_1_3_5_1"><i><font color="Green" title="E38">E2</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3"><i><font color="Green" title="E30">C1</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3"><i><font color="Green" title="E31">C2</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3_5_1"><i><font color="Green" title="E35">e0</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E7:27_1_1_1_3_5_1"><i><font color="Green" title="E39">E3</font></i></a>, <a class="txt" href="#E5:27_1_1_1_3"><i><font color="Green" title="E33">C4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_2"/><i><font color="Green" title="E34">D1</font></i>: 
not <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_5_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_3_5_2_1_1"/>
( <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_2_1_1_1"/><i><font color="Green" title="E35">F0</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_2_1_1_1"/><b>then </b><i><font color="Green" title="E36">G0</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:27_1_1_1_3_5_2_1_1_1"/><i><font color="Green" title="E37">G1</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</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> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1_1_3_5_2_1_1_1_1"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="ref" href="xboole_1.html#T29" title="XBOOLE_1:th.29">XBOOLE_1:29</a></i>;<br/>
<b>hence </b><a NAME="E2:27_1_1_1_3_5_2_1_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</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="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E4:27_1_1_1_3_5_2_1_1_1"/><i><font color="Green" title="E38">G2</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E35">F0</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E5:27_1_1_1_3_5_2_1_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:27_1_1_1_3_5_2_1_1_1_2_1"/>
( <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_2_1_1_1_2_1_1"/><i><font color="Green" title="E39">H0</font></i>: 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_2_1_1_1_2_1_1"/>
<font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_1_2_1_1"><i><font color="Green" title="E39">H0</font></i></a></i>;<br/><a NAME="E3:27_1_1_1_3_5_2_1_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E40">j1</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E38">G2</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="E4:27_1_1_1_3_5_2_1_1_1_2_1_1"/>
( <font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">E</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c9">f</font> &amp; <font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E'</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">f'</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T40" title="FUNCT_2:th.40">FUNCT_2:40</a></i>;<br/><b>hence </b><a NAME="E5:27_1_1_1_3_5_2_1_1_1_2_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_5_2_1_1_1_2_1_1"><i><font color="Green" title="E40">j1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3"><i><font color="Green" title="E29">C0</font></i></a>, <a class="txt" href="#E5:27_1_1_1_3"><i><font color="Green" title="E33">C4</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E36">G0</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_2_1_1_1_2_1_2"/><i><font color="Green" title="E39">H1</font></i>: 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_2_1_1_1_2_1_2"/><i><font color="Green" title="E40">J0</font></i>: 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E3:27_1_1_1_3_5_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:27_1_1_1_3_5_2_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E40">J0</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:27_1_1_1_3_5_2_1_1_1_2_1_2"/><b>then </b>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E36">G0</font></i></a>, <a class="txt" href="#E5:27_1_1_1_3"><i><font color="Green" title="E33">C4</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E37">G1</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E39">H1</font></i></a></i>;<br/><a NAME="E6:27_1_1_1_3_5_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E41">J2</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3"><i><font color="Green" title="E29">C0</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3"><i><font color="Green" title="E31">C2</font></i></a>, <a class="ref" href="funct_2.html#T40" title="FUNCT_2:th.40">FUNCT_2:40</a>, <a class="txt" href="#E4:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E38">G2</font></i></a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><a NAME="E7:27_1_1_1_3_5_2_1_1_1_2_1_2"/><i><font color="Green" title="E42">J3</font></i>: 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E38">G2</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E8:27_1_1_1_3_5_2_1_1_1_2_1_2"/><i><font color="Green" title="E43">J4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_1"><i><font color="Green" title="E35">F0</font></i></a></i>;<br/><a NAME="E9:27_1_1_1_3_5_2_1_1_1_2_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E44">J5</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:27_1_1_1_3"><i><font color="Green" title="E32">C3</font></i></a>, <a class="txt" href="#E7:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E42">J3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E8:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E43">J4</font></i></a></i>
;<br/><a NAME="E11:27_1_1_1_3_5_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E12:27_1_1_1_3_5_2_1_1_1_2_1_2"/><b>then </b><i><font color="Green" title="E45">j6</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E13:27_1_1_1_3_5_2_1_1_1_2_1_2"/>
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E14:27_1_1_1_3_5_2_1_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E15:27_1_1_1_3_5_2_1_1_1_2_1_2"/><b>then </b>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E44">J5</font></i></a>, <a class="txt" href="#E12:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E45">j6</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E39">H1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a></i>;<br/><b>hence </b><a NAME="E16:27_1_1_1_3_5_2_1_1_1_2_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:27_1_1_1_3_5_2_1_1_1_2_1_2"><i><font color="Green" title="E41">J2</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E6:27_1_1_1_3_5_2_1_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_2_1_1_2"/><i><font color="Green" title="E35">F1</font></i>: 
not <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_2_1_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:27_1_1_1_3_5_2_1_1_2_1"/><i><font color="Green" title="E36">g0</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T29" title="XBOOLE_1:th.29">XBOOLE_1:29</a></i>;<br/>
<a NAME="E2:27_1_1_1_3_5_2_1_1_2_1"/><i><font color="Green" title="E37">G1</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E3:27_1_1_1_3_5_2_1_1_2_1"/>
<font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 
<b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:27_1_1_1_3_5_2_1_1_2_1"/><b>then </b><i><font color="Green" title="E38">G2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</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="#E1:27_1_1_1_3_5_2_1_1_2"><i><font color="Green" title="E35">F1</font></i></a>, <a class="ref" href="kolmog01.html#T1" target="_self" title="KOLMOG01:th.1">Th12</a>, <a class="txt" href="#E2:27_1_1_1_3_5_2_1_1_2_1"><i><font color="Green" title="E37">G1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_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="E1:27_1_1_1_3_5_2_1_1_2_1_1"/>
( <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_2_1_1_2_1_1_1"/><i><font color="Green" title="E39">H0</font></i>: 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_2_1_1_2_1_1_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then </b><i><font color="Green" title="E40">J0</font></i>:  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:27_1_1_1_3"><i><font color="Green" title="E33">C4</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1"><i><font color="Green" title="E36">g0</font></i></a>, <a class="txt" href="#E4:27_1_1_1_3_5_2_1_1_2_1"><i><font color="Green" title="E38">G2</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a>, <a class="txt" href="#E1:27_1_1_1_3"><i><font color="Green" title="E29">C0</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3"><i><font color="Green" title="E30">C1</font></i></a></i>
;<br/><a NAME="E4:27_1_1_1_3_5_2_1_1_2_1_1_1"/>
( <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> &amp; <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font> )
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E39">H0</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E5:27_1_1_1_3_5_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E41">J1</font></i>: 
<font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T40" title="FUNCT_2:th.40">FUNCT_2:40</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></i>;<br/><a NAME="E6:27_1_1_1_3_5_2_1_1_2_1_1_1"/><i><font color="Green" title="E42">J2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E39">H0</font></i></a></i>;<br/><a NAME="E7:27_1_1_1_3_5_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E39">H0</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E8:27_1_1_1_3_5_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>;<br/><a NAME="E9:27_1_1_1_3_5_2_1_1_2_1_1_1"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E43">J3</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:27_1_1_1_3"><i><font color="Green" title="E32">C3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E6:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E42">J2</font></i></a></i>
;<br/><a NAME="E11:27_1_1_1_3_5_2_1_1_2_1_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E12:27_1_1_1_3_5_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E44">J4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</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="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a>, <a class="ref" href="kolmog01.html#T1" target="_self" title="KOLMOG01:th.1">Th12</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E13:27_1_1_1_3_5_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E14:27_1_1_1_3_5_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E15:27_1_1_1_3_5_2_1_1_2_1_1_1"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E40">J0</font></i></a>, <a class="txt" href="#E5:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E41">J1</font></i></a>, <a class="txt" href="#E10:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E43">J3</font></i></a>, <a class="txt" href="#E12:27_1_1_1_3_5_2_1_1_2_1_1_1"><i><font color="Green" title="E44">J4</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2"><i><font color="Green" title="E35">F1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1_1_3_5_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:27_1_1_1_3_5_2_1_1_2_1_1_2"/><i><font color="Green" title="E39">H1</font></i>: 
not <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:27_1_1_1_3_5_2_1_1_2_1_1_2"/><i><font color="Green" title="E40">J1</font></i>: 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E3:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E40">J1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then </b><i><font color="Green" title="E41">J2</font></i>:  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:27_1_1_1_3"><i><font color="Green" title="E33">C4</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1"><i><font color="Green" title="E36">g0</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E39">H1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1"><i><font color="Green" title="E36">g0</font></i></a>, <a class="txt" href="#E4:27_1_1_1_3_5_2_1_1_2_1"><i><font color="Green" title="E38">G2</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a>, <a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">B4</font></i></a></i>
;<br/><a NAME="E6:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E7:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E8:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E42">J3</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_3"><i><font color="Green" title="E32">C3</font></i></a></i>;<br/><i><font color="Green" title="E43">J4</font></i>: <font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">E</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font>

;<br/><a NAME="E10:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E11:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E12:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E13:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E44">J5</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E42">J3</font></i></a>, <a class="txt" href="#E9:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E43">J4</font></i></a>, <a class="ref" href="xboole_1.html#T47" title="XBOOLE_1:th.47">XBOOLE_1:47</a></i>;<br/><a NAME="E14:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E15:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E45">j6</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E16:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E17:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E18:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E46">J7</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E13:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E44">J5</font></i></a>, <a class="txt" href="#E15:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E45">j6</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2"><i><font color="Green" title="E35">F1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a></i>;<br/><a NAME="E19:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E20:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E47">J8</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green" title="E48">J9</font></i>: <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">E</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font>

;<br/><a NAME="E22:27_1_1_1_3_5_2_1_1_2_1_1_2"/><i><font color="Green" title="E49">J10</font></i>: 
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T47" title="XBOOLE_1:th.47">XBOOLE_1:47</a></i>;<br/><a NAME="E23:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E24:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><a NAME="E25:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E50">J11</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E20:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E47">J8</font></i></a>, <a class="txt" href="#E21:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E48">J9</font></i></a>, <a class="txt" href="#E22:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E49">J10</font></i></a>, <a class="txt" href="#E4:27_1_1_1_3"><i><font color="Green" title="E32">C3</font></i></a></i>;<br/><a NAME="E26:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/><a NAME="E27:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E51">j13</font></i>: 
<font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E28:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">E'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><a NAME="E29:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f'</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E30:27_1_1_1_3_5_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E52">J14</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E25:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E50">J11</font></i></a>, <a class="txt" href="#E27:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E51">j13</font></i></a>, <a class="ref" href="setfam_1.html#T10" title="SETFAM_1:th.10">SETFAM_1:10</a>, <a class="txt" href="#E1:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E39">H1</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_5_2"><i><font color="Green" title="E34">D1</font></i></a></i>;<br/><a NAME="E31:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <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"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p4">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p4">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">E</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c10">E'</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">f'</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">E'</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">E</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3"><i><font color="Green" title="E29">C0</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3"><i><font color="Green" title="E30">C1</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3"><i><font color="Green" title="E31">C2</font></i></a>, <a class="txt" href="#E5:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E41">J2</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="E32:27_1_1_1_3_5_2_1_1_2_1_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E18:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E46">J7</font></i></a>, <a class="txt" href="#E30:27_1_1_1_3_5_2_1_1_2_1_1_2"><i><font color="Green" title="E52">J14</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:27_1_1_1_3_5_2_1_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E3:27_1_1_1_3_5_2"/>
<font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">g</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1"><i><font color="Green" title="E22">B1</font></i></a>, <a class="txt" href="#E5:27_1_1_1"><i><font color="Green" title="E23">B2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E15:27_1_1_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> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Omega</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span> implies <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Omega</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_4"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Omega</font></span><br/>

<a NAME="E2:27_1_1_1_4"/><b>then </b><i><font color="Green" title="E29">D0</font></i>: 
( <font color="Maroon" title="c15">y</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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font></span>)</span> &amp; <font color="Maroon" title="c15">y</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="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">S</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:27_1_1_1_4"/><i><font color="Green" title="E30">D1</font></i>: 
<font color="Maroon" title="c16">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">S'</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:27_1_1_1_4"/><i><font color="Green" title="E31">D2</font></i>: 
<font color="Maroon" title="c17">S'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E7:27_1_1_1_4"/><i><font color="Green" title="E32">D3</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">S</font> &amp; <font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">S'</font> )
 
<b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a>, <a class="txt" href="#E2:27_1_1_1_4"><i><font color="Green" title="E29">D0</font></i></a>, <a class="txt" href="#E4:27_1_1_1_4"><i><font color="Green" title="E30">D1</font></i></a>, <a class="txt" href="#E6:27_1_1_1_4"><i><font color="Green" title="E31">D2</font></i></a></i>;<br/>
<a NAME="E8:27_1_1_1_4"/><i><font color="Green" title="E33">D4</font></i>: 
<font color="Maroon" title="c16">S</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">S'</font> is    <a href="prob_1.html#M2" title="PROB_1:mode.2">Event</a> of <font color="Maroon" title="c3">Sigma</font>
 
<b>by </b><i><a class="ref" href="prob_1.html#T49" title="PROB_1:th.49">PROB_1:49</a>, <a class="txt" href="#E4:27_1_1_1_4"><i><font color="Green" title="E30">D1</font></i></a>, <a class="txt" href="#E6:27_1_1_1_4"><i><font color="Green" title="E31">D2</font></i></a></i>;<br/>
<a NAME="E9:27_1_1_1_4"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">S</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c17">S'</font>
 
<b>by </b><i><a class="txt" href="#E7:27_1_1_1_4"><i><font color="Green" title="E32">D3</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="E10:27_1_1_1_4"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Omega</font>
 <b>by </b><i><a class="txt" href="#E8:27_1_1_1_4"><i><font color="Green" title="E33">D4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c15">xX</font> = <font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Omega</font> <b>by </b><i><a class="txt" href="#E3:27_1_1_1"><i><font color="Green" title="E22">B1</font></i></a>, <a class="txt" href="#E5:27_1_1_1"><i><font color="Green" title="E23">B2</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E17:27_1_1_1"/>
<font color="Maroon" title="c15">xX</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="#E13:27_1_1_1"><i><font color="Green" title="E27">B6</font></i></a>, <a class="txt" href="#E14:27_1_1_1"><i><font color="Green" title="E28">B7</font></i></a>, <a class="ref" href="kolmog01.html#D9" target="_self" title="KOLMOG01:def.9">Def9</a></i>;<br/>
<b>hence </b><a NAME="E18:27_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:27_1_1"/>
 <a href="kolmog01.html#K7" title="KOLMOG01:func.7">MeetSections</a> <font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c4">F</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">intersection_stable</a>
 <b>by </b><i><a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
