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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#V1" title="INTERVA1:attr.1">ordered</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> holds   <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span>)</span>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="interva1.html#V1" title="INTERVA1:attr.1">ordered</a>  <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font></span><br/>



<a NAME="E1:47"/><span class="lab"><font color="Green" title="E36">a0</font></span>: 
 <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">B1</font> =  <a href="interva1.html#K7" title="INTERVA1:func.7">min</a> <font color="Maroon" title="c3">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B2</font> =  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c3">B</font>;<br/>
<span class="kw">assume </span><a NAME="E1:47_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">Y</font>, <font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:47_1"/><span class="lab"><font color="Green" title="E37">a2</font></span>: 
( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Y</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c8">Z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D4" title="SETFAM_1:def.4">SETFAM_1:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">Y1</font>, <font color="Maroon" title="c10">Y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:47_1"/><span class="lab"><font color="Green" title="E38">a3</font></span>: 
( <font color="Maroon" title="c9">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">Y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">Y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a>, <a class="txt" href="#E3:47_1"><span class="lab"><font color="Green" title="E37">a2</font></span></a></span>;<br/>
<a NAME="E6:47_1"/>
(  <a href="interva1.html#K7" title="INTERVA1:func.7">min</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y2</font> &amp; <font color="Maroon" title="c10">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:47_1"><span class="lab"><font color="Green" title="E38">a3</font></span></a>, <a class="ref" href="interva1.html#T28" target="_self" title="INTERVA1:th.28">Nowe1</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E7:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">a5</font></span>: 
<font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c10">Y2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E8:47_1"/>
<font color="Maroon" title="c8">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1"><span class="lab"><font color="Green" title="E37">a2</font></span></a>, <a class="ref" href="interva1.html#T28" target="_self" title="INTERVA1:th.28">Nowe1</a></span>;<br/>
<a NAME="E9:47_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">a6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1"><span class="lab"><font color="Green" title="E37">a2</font></span></a>, <a class="txt" href="#E5:47_1"><span class="lab"><font color="Green" title="E38">a3</font></span></a>, <a class="txt" href="#E7:47_1"><span class="lab"><font color="Green" title="E39">a5</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/>
<a NAME="E10:47_1"/>
(  <a href="interva1.html#K7" title="INTERVA1:func.7">min</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c8">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1"><span class="lab"><font color="Green" title="E37">a2</font></span></a>, <a class="ref" href="interva1.html#T28" target="_self" title="INTERVA1:th.28">Nowe1</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E11:47_1"/><span class="kw">then </span>
(  <a href="interva1.html#K7" title="INTERVA1:func.7">min</a> <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c4">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c3">B</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:47_1"><span class="lab"><font color="Green" title="E40">a6</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:47_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="interva1.html#T28" target="_self" title="INTERVA1:th.28">Nowe1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:47"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">B1</font> =  <a href="interva1.html#K7" title="INTERVA1:func.7">min</a> <font color="Maroon" title="c3">B</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">B2</font> =  <a href="interva1.html#K8" title="INTERVA1:func.8">max</a> <font color="Maroon" title="c3">B</font>;<br/>
<span class="kw">assume </span><a NAME="E1:47_2"/><span class="lab"><font color="Green" title="E37">a1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">b</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font>;<br/>
<a NAME="E3:47_2"/><span class="lab"><font color="Green" title="E38">L1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c7">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T22" title="XBOOLE_1:th.22">XBOOLE_1:22</a></span>;<br/>
<a NAME="E4:47_2"/>
<font color="Maroon" title="c7">b</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E37">a1</font></span></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:47_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_2"><span class="lab"><font color="Green" title="E38">L1</font></span></a>, <a class="txt" href="#E1:47_2"><span class="lab"><font color="Green" title="E37">a1</font></span></a>, <a class="ref" href="setfam_1.html#D4" title="SETFAM_1:def.4">SETFAM_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:47"/>
 <a href="setfam_1.html#K2" title="SETFAM_1:func.2">UNION</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3" title="SETFAM_1:func.3">INTERSECTION</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span>)</span>,<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>, <a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E36">a0</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
