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

<b>let </b><font color="Maroon" title="c1">Y</font>, <font color="Maroon" title="c2">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">F</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">Y</font><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="c2">X</font>,<font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">F</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <font color="Olive" title="b1">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Olive" title="b1">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/>for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> holds  <font color="Olive" title="b1">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font>  st <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> holds <br/>for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> holds  <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">b</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> implies for <font color="Olive" title="b1">a</font>, <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="c2">X</font> holds  <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V3" title="BINOP_1:attr.3">idempotent</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</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>, <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="c2">X</font> holds  <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><br/>

<b>consider </b><font color="Maroon" title="c7">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">Y</font><b> such that </b><br/><a NAME="E3:31"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>and </b><br/><a NAME="E4:31"/>
for <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">Y</font>  st <font color="Olive" title="b1">e</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font>
 <b>and </b><br/><a NAME="E5:31"/><i><font color="Green" title="E23">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> holds  <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 <b>and </b><br/><a NAME="E6:31"/><i><font color="Green" title="E24">A4</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="c2">X</font>  st <font color="Olive" title="b1">B'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <font color="Olive" title="b1">B'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">B'</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">x</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> <font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">B'</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:31"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="setwiseo.html#T25" target="_self" title="SETWISEO:th.25">Th25</a></i>;<br/>
<a NAME="E7:31"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="setwiseo.html#K3" title="SETWISEO:func.3">{.</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span><a href="setwiseo.html#K3" title="SETWISEO:func.3">.}</a></span>,<font color="Maroon" title="c4">f</font> = 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:31"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:31"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="#E7:31"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T12" title="ZFMISC_1:th.12">ZFMISC_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c3">F</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:31"><i><font color="Green" title="E23">A3</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>
