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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="coh_sp.html#K21" title="COH_SP:func.21">MapsT</a> <font color="Maroon" title="c1">X</font> holds <br/> ( <font color="Olive" title="b1">m</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">m</font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="coh_sp.html#K21" title="COH_SP:func.21">MapsT</a> <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">m</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">i2</font> =  <a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span>;<br/>
<a NAME="E1:77"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a></span>)</span> &amp;  <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">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="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T46" target="_self" title="COH_SP:th.46">Th47</a></span>;<br/>
<a NAME="E2:77"/><span class="lab"><font color="Green" title="E46">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"><span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T43" target="_self" title="COH_SP:th.43">Th44</a></span>;<br/>
<a NAME="E3:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">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> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> 
 
<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="E4:77"/>
( <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <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 <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a>  <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="coh_sp.html#T43" target="_self" title="COH_SP:th.43">Th44</a></span>;<br/>
<a NAME="E5:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">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> <span class="p1">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:77"><span class="lab"><font color="Green" title="E46">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="E6:77"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
 <a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">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="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T46" target="_self" title="COH_SP:th.46">Th47</a></span>;<br/>
<a NAME="E7:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 <a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">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="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E8:77"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">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="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">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="#E6:77"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:77"/>
<font color="Maroon" title="c2">m</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:77"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E7:77"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="txt" href="coh_sp.html#E67"><span class="lab"><font color="Green" title="E38">Lm3</font></span></a>, <a class="ref" href="relat_1.html#T52" title="RELAT_1:th.52">RELAT_1:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="E10:77"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
( <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="coh_sp.html#K18" title="COH_SP:func.18">`2</a></span>)</span> &amp;  <a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">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="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T46" target="_self" title="COH_SP:th.46">Th47</a></span>;<br/>
<a NAME="E11:77"/><span class="lab"><font color="Green" title="E52">A8</font></span>: 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">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="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#T46" target="_self" title="COH_SP:th.46">Th47</a></span>;<br/>
<a NAME="E12:77"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A9</font></span>: 
 <a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">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="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E13:77"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">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="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p3">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:77"><span class="lab"><font color="Green" title="E52">A8</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:77"/>
<span class="p1">(<span class="default"><a href="coh_sp.html#K24" title="COH_SP:func.24">id$</a> <span class="p2">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="#E3:77"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E10:77"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E12:77"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="coh_sp.html#E67"><span class="lab"><font color="Green" title="E38">Lm3</font></span></a>, <a class="ref" href="relat_1.html#T53" title="RELAT_1:th.53">RELAT_1:53</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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