<?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">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="coh_sp.html#K21" title="COH_SP:func.21">MapsT</a> <font color="Maroon" title="c1">X</font>  st  <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Olive" title="b1">m2</font> <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="Olive" title="b2">m1</font> &amp;  <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Olive" title="b3">m3</font> <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="Olive" title="b1">m2</font> holds <br/><font color="Olive" title="b3">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Olive" title="b1">m2</font></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Olive" title="b2">m1</font></span><br/><span class="kw">let </span><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="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"> (  <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m2</font> <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="c3">m1</font> &amp;  <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c4">m3</font> <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">m2</font> implies <font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:75"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c2">m2</font> <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="c3">m1</font>
 <span class="kw">and </span><br/><a NAME="E2:75"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <font color="Maroon" title="c4">m3</font> <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">m2</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="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</font></span><br/>

<a NAME="E3:75"/><span class="lab"><font color="Green" title="E45">A3</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">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</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> <font color="Maroon" title="c2">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E4:75"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E5:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E6:75"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</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">m2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E7:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A6</font></span>: 
 <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"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</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="c3">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E8:75"/>
 <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</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="c3">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E9:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</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="c3">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E10:75"/>
 <a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</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> <font color="Maroon" title="c4">m3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E11:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A8</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"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</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> <font color="Maroon" title="c4">m3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E12:75"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E13:75"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:75"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E6:75"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="ref" href="coh_sp.html#T44" target="_self" title="COH_SP:th.44">Th45</a></span>;<br/>
<a NAME="E14:75"/>
 <a href="coh_sp.html#K23" title="COH_SP:func.23">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</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="c4">m3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:75"><span class="lab"><font color="Green" title="E44">A2</font></span></a>, <a class="txt" href="#E3:75"><span class="lab"><font color="Green" title="E45">A3</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="E15:75"/>
<font color="Maroon" title="c4">m3</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m2</font> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</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="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c2">m2</font></span>)</span> <a href="coh_sp.html#K25" title="COH_SP:func.25">*</a> <font color="Maroon" title="c3">m1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:75"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="txt" href="#E9:75"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E13:75"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E7:75"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E11:75"><span class="lab"><font color="Green" title="E50">A8</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#T36" title="RELAT_1:th.36">RELAT_1:36</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/>


</div>
