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

<b>let </b><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">a1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font>;<br/><b>let </b><font color="Maroon" title="c4">a2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font>;<br/><b>let </b><font color="Maroon" title="c5">f1</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">D1</font>;<br/><b>let </b><font color="Maroon" title="c6">f2</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D2</font>;<br/>











<b>thus </b><a NAME="E1:39"/>
( <font color="Maroon" title="c3">a1</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c5">f1</font> &amp; <font color="Maroon" title="c4">a2</font> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <font color="Maroon" title="c6">f2</font> implies <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="binop_1.html#R1" title="BINOP_1:pred.1">is_a_left_unity_wrt</a> <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">b1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font> holds  <font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><br/>

<b>assume </b><a NAME="E2:39_1"/><i><font color="Green" title="E28">A2</font></i>: 
for <font color="Olive" title="b1">b2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font> holds  <font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">a2</font>,<font color="Olive" title="b1">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><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><span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1;<br/>
<div><i><font color="Green" title="E29">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">b1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font>;<br/><b>let </b><font color="Maroon" title="c8">b2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font>;<br/><span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c7">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c8">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c8">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E1:39_1"><i><font color="Green" title="E27">A1</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E2:39_1"><i><font color="Green" title="E28">A2</font></i></a></i>
;<br/><b>hence </b><a NAME="E2:39_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E4:39_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="filter_1.html#S4" target="_self" title="FILTER_1:sch.4">FILTER_1:sch 4</a>(<a class="txt" href="filter_1.html#E3:39_1"><i><font color="Green" title="E29">A3</font></i></a>);</i> <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><br/>


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

<b>assume </b><a NAME="E2:39"/><i><font color="Green" title="E27">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds  <span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><br/>

<b>thus </b><a NAME="E3:39"/>
for <font color="Olive" title="b1">b1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font> holds  <font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Olive" title="b1">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font>;<br/>

<b>consider </b><font color="Maroon" title="c8">b2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font>;<br/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c7">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c8">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E2:39"><i><font color="Green" title="E27">A4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E3:39_2"/>
<font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c7">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b1</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c7">b2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D16" title="BINOP_1:def.16">BINOP_1:def 16</a><br/>

<b>consider </b><font color="Maroon" title="c8">b1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D1</font>;<br/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c8">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c7">b2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> = 
<span class="p1"><a href="filter_1.html#K6" title="FILTER_1:func.6">|:</a><span class="default"><font color="Maroon" title="c5">f1</font>,<font color="Maroon" title="c6">f2</font></span><a href="filter_1.html#K6" title="FILTER_1:func.6">:|</a></span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c7">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="filter_1.html#T22" target="_self" title="FILTER_1:th.22">Th22</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">b1</font>,<font color="Maroon" title="c7">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="txt" href="filter_1.html#E2:39"><i><font color="Green" title="E27">A4</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:39"/>
<font color="Maroon" title="c6">f2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>


</div>
