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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#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:101_1_1_1"/>
<font color="Maroon" title="c3">x</font> <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#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> :  ex <font color="Olive" title="b2">H</font> being  <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/><a NAME="E2:101_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#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> st <br/>( <font color="Maroon" title="c3">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="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="E3:101_1_1_1"/>
<font color="Maroon" title="c3">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#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:101_1_1"/>
<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#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span> :  ex <font color="Olive" title="b2">H</font> being  <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>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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#K28" title="GRCAT_1:func.28">GroupCat</a> <font color="Maroon" title="c1">UN</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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