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<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" target="_self">AbGroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" target="_self">AbGroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being <a href="midsp_2.html#V1">midpoint_operator</a> <a href="vectsp_1.html#NM1">AbGroup</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="algstr_1.html#K2">L_Trivial</a> ;<br/>
<a NAME="E1:109_1_1"/>
 <a href="algstr_1.html#K2">L_Trivial</a>  <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" target="_self">AbGroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="grcat_1.html#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E2:109_1_1"/><b>then </b>
 <a href="algstr_1.html#K2">L_Trivial</a>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" target="_self">AbGroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being <a href="midsp_2.html#V1">midpoint_operator</a> <a href="vectsp_1.html#NM1">AbGroup</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" target="_self">AbGroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> : ex <font color="Olive">b<sub>1</sub></font> being <a href="midsp_2.html#V1">midpoint_operator</a> <a href="vectsp_1.html#NM1">AbGroup</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:109_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:109_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" target="_self">AbGroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being <a href="midsp_2.html#V1">midpoint_operator</a> <a href="vectsp_1.html#NM1">AbGroup</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><b>hence </b><a NAME="E3:109_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="cat_1.html#U1">Objects</a> of <span class="p1">(<span class="default"><a href="grcat_1.html#K30" target="_self">AbGroupCat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:109_1_1"/>
not  <a href="grcat_1.html#K31" target="_self">MidOpGroupObjects</a> <font color="Maroon">c<sub>1</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


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