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

<span class="kw">set </span><font color="Maroon" title="c2">T</font> =  <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">D2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">D1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> :  ex <font color="Olive" title="b2">H</font> being   <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> </span> } </span> ;<br/>
<a NAME="E1:102_1_1"/>
 <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="grcat_1.html#T37" target="_self" title="GRCAT_1:th.37">Th51</a></span>;<br/>
<a NAME="E2:102_1_1"/><span class="kw">then </span>
 <a href="algstr_0.html#K3" title="ALGSTR_0:func.3">Trivial-addLoopStr</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> :  ex <font color="Olive" title="b2">H</font> being   <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> </span> } </span> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">D1</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> :  ex <font color="Olive" title="b2">H</font> being   <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> </span> } </span>  as   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>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">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"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D1</font> implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:102_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">D1</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span></span><br/><a NAME="E2:102_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">G</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> st <br/>( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp;  ex <font color="Olive" title="b2">H</font> being   <a href="midsp_2.html#V1" title="MIDSP_2:attr.1">midpoint_operator</a>  <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">AbGroup</a> st <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">H</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:102_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K29" title="GRCAT_1:func.29">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:102_1_1"/>
 not  <a href="grcat_1.html#K30" title="GRCAT_1:func.30">MidOpGroupObjects</a> <font color="Maroon" title="c1">UN</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <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>
