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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</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> holds <br/> ( <font color="Olive" title="b1">m</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">i1</font> =  <a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">i2</font> =  <a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<a NAME="E1:25"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</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="c2">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T11" target="_self" title="ENS_1:th.11">Th11</a></span>;<br/>
<a NAME="E2:25"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c2">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span>,<span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T9" target="_self" title="ENS_1:th.9">Th9</a></span>;<br/>
<a NAME="E3:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E4:25"/>
(  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</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>  or  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T9" target="_self" title="ENS_1:th.9">Th9</a></span>;<br/>
<a NAME="E5:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </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">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:25"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</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="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T11" target="_self" title="ENS_1:th.11">Th11</a></span>;<br/>
<a NAME="E7:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A6</font></span>: 
 <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></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="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<a NAME="E8:25"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <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">m</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:25"/>
<font color="Maroon" title="c2">m</font> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="txt" href="ens_1.html#E15"><span class="lab"><font color="Green" title="E8">Lm2</font></span></a>, <a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font></span><br/>

<a NAME="E10:25"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
( <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</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="c2">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T11" target="_self" title="ENS_1:th.11">Th11</a></span>;<br/>
<a NAME="E11:25"/><span class="lab"><font color="Green" title="E24">A8</font></span>: 
 <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</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="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T11" target="_self" title="ENS_1:th.11">Th11</a></span>;<br/>
<a NAME="E12:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A9</font></span>: 
 <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"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<a NAME="E13:25"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <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">m</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"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m</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">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:25"/>
<span class="p1">(<span class="default"><a href="ens_1.html#K5" title="ENS_1:func.5">id$</a> <span class="p2">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E10:25"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="ens_1.html#E15"><span class="lab"><font color="Green" title="E8">Lm2</font></span></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</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>
