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<b>set </b><font color="Maroon" title="c2">D2</font> = the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" title="GRCAT_1:func.30">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c3">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="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" title="GRCAT_1:func.30">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#NM1" title="VECTSP_1:NM.1">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/>
<b>set </b><font color="Maroon" title="c4">T</font> =  <a href="algstr_1.html#K2" title="ALGSTR_1:func.2">L_Trivial</a> ;<br/>
<a NAME="E1:107_1_1"/>
 <a href="algstr_1.html#K2" title="ALGSTR_1:func.2">L_Trivial</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" title="GRCAT_1:func.30">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T51" target="_self" title="GRCAT_1:th.51">Th51</a></i>;<br/>
<a NAME="E2:107_1_1"/><b>then </b>
 <a href="algstr_1.html#K2" title="ALGSTR_1:func.2">L_Trivial</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="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" title="GRCAT_1:func.30">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#NM1" title="VECTSP_1:NM.1">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/>
<b>then reconsider </b><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="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" title="GRCAT_1:func.30">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#NM1" title="VECTSP_1:NM.1">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="107_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:107_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>
 ;<br/><a NAME="E2:107_1_1_1"/><b>then </b>
 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="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" title="GRCAT_1:func.30">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#NM1" title="VECTSP_1:NM.1">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/><b>hence </b><a NAME="E3:107_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="cat_1.html#U1" title="CAT_1:sel.1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" title="GRCAT_1:func.30">AbGroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:107_1_1"/>
not  <a href="grcat_1.html#K31" title="GRCAT_1:func.31">MidOpGroupObjects</a> <font color="Maroon" title="c1">UN</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>


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