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

<b>set </b><font color="Maroon" title="c1">G</font> =  <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #);<br/>
<a NAME="E1:6"/><i><font color="Green" title="E7">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #)  ex <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #) st  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #);<br/>

<b>consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #);<br/>
<b>take </b>
<font color="Maroon" title="c3">x</font>
;<br/>

<b>thus </b> <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Maroon" title="c3">x</font> = 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:6"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #)  st  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #) holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #)
 
<b>by </b><i><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:6"/>
 <a href="algstr_0.html#G2" title="ALGSTR_0:aggr.2">addLoopStr</a>(# 1,<a href="funct_5.html#K9" title="FUNCT_5:func.9">op2</a> ,<a href="funct_5.html#K5" title="FUNCT_5:func.5">op0</a>  #) is <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>
 <b>by </b><i><a class="txt" href="grcat_1.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="midsp_2.html#D5" title="MIDSP_2:def.5">MIDSP_2:def 5</a></i>;<br/>


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