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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">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="c1">X</font><br/> for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Olive" title="b3">B</font>,<font color="Olive" title="b2">f</font> iff  ex <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">i</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">A</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">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">A</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  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Olive" title="b2">B</font>,<font color="Olive" title="b1">f</font> iff  ex <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>, <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">A</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  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Olive" title="b1">B</font>,<font color="Maroon" title="c3">f</font> iff  ex <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">i</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">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="c1">X</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">f</font> iff  ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">i</font> ) )</span><br/><b>let </b><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">f</font> iff  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> ) )</span><br/>









<div><i><font color="Green" title="E49">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">i'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i'</font> )</span><br/><b>then consider </b><font color="Maroon" title="c6">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:75_1"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>and </b><br/><a NAME="E4:75_1"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><a NAME="E5:75_1"/>
<font color="Maroon" title="c3">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">A</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E77"><i><font color="Green" title="E47">Lm2</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">Z</font> = <font color="Maroon" title="c6">Z</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="c2">A</font> <b>by </b><i><a class="txt" href="setwiseo.html#E4:75_1"><i><font color="Green" title="E51">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c8">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E8:75_1"/><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c7">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">i</font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E4:75_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/><b>take </b><font color="Maroon" title="c9">i'</font> = <font color="Maroon" title="c8">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i'</font> )</span><br/><b>thus </b><a NAME="E9:75_1"/>
( <font color="Maroon" title="c9">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">i'</font> )
 <b>by </b><i><a class="txt" href="setwiseo.html#E3:75_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="setwiseo.html#E8:75_1"><i><font color="Green" title="E52">A4</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="hs2(this)" href="javascript:()" title="75_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c6">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E2:75_2"/><i><font color="Green" title="E50">A5</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span></span><br/><a NAME="E3:75_2"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="txt" href="setwiseo.html#E2:75_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><b>hence </b><a NAME="E4:75_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="setwiseo.html#K8" title="SETWISEO:func.8">.:</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="setwiseo.html#E2:75_2"><i><font color="Green" title="E50">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 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>hence </b><a NAME="E3:75"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setwiseo.html#K10" title="SETWISEO:func.10">FinUnion</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">f</font> iff  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font> ) )
 <b>by </b><i><a class="txt" href="setwiseo.html#E1:75"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="setwiseo.html#T61" target="_self" title="SETWISEO:th.61">Th61</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>
