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

<b>let </b><font color="Maroon" title="c1">V</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">m2</font>, <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>  st  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b2">m1</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b3">m3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b1">m2</font> holds <br/><font color="Olive" title="b3">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Olive" title="b2">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Olive" title="b1">m2</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Olive" title="b2">m1</font></span><br/><b>let </b><font color="Maroon" title="c2">m2</font>, <font color="Maroon" title="c3">m1</font>, <font color="Maroon" title="c4">m3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c3">m1</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c4">m3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m2</font> implies <font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c3">m1</font>
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c4">m3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span><br/>

<a NAME="E3:24"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c3">m1</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m2</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c4">m3</font> )
 
<b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E2:24"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></i>;<br/>
<a NAME="E4:24"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c3">m1</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c4">m3</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c3">m1</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c4">m3</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:24"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E2:24"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>;<br/>
<b>hence </b><a NAME="E5:24"/>
<font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c3">m1</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E15"><i><font color="Green" title="E8">Lm2</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>
