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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E145">Th146</font></span>: <a NAME="T192"><span class="comment"><font color="firebrick">:: GROUP_1A:192</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">G</font> being   <a href="group_1a.html#NM1" title="GROUP_1A:NM.1">addGroup</a><br/>  for <font color="Olive" title="b2">H</font> being    <a href="group_1a.html#M1" title="GROUP_1A:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font>  st  <a href="group_1a.html#K20" title="GROUP_1A:func.20">Left_Cosets</a> <font color="Olive" title="b2">H</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  holds <br/>(  ex <font color="Olive" title="b3">B</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1a.html#K20" title="GROUP_1A:func.20">Left_Cosets</a> <font color="Olive" title="b2">H</font> &amp;  <a href="group_1a.html#K23" title="GROUP_1A:func.23">index</a> <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b3">B</font> ) &amp;  ex <font color="Olive" title="b3">C</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1a.html#K21" title="GROUP_1A:func.21">Right_Cosets</a> <font color="Olive" title="b2">H</font> &amp;  <a href="group_1a.html#K23" title="GROUP_1A:func.23">index</a> <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b3">C</font> ) )</div></div>
