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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="bspace.html#K7" title="BSPACE:func.7">bspace</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">C</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">C</font> )
 ;<br/>

<b>thus </b><a NAME="E2:19"/>
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c7">C</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<a NAME="E1:19_1"/>
<font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c7">C</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="bspace.html#D5" target="_self" title="BSPACE:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1"/>
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c7">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="bspace.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="bspace.html#D5" target="_self" title="BSPACE:def.5">Def5</a></i>;<br/>


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

<b>thus </b><a NAME="E3:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c7">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<a NAME="E1:19_2"/>
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c6">B</font>
 
<b>by </b><i><a class="txt" href="bspace.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="bspace.html#D5" target="_self" title="BSPACE:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E2:19_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="subset_1.html#K5" title="SUBSET_1:func.5">\+\</a> <font color="Maroon" title="c7">C</font>
 <b>by </b><i><a class="txt" href="bspace.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="bspace.html#D5" target="_self" title="BSPACE:def.5">Def5</a></i>;<br/>


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


</div>
