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<div><span class="kw">theorem </span><a NAME="T63"><span class="comment"><font color="firebrick">:: GROUP_24:63</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">H</font>, <font color="Olive" title="b2">G</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b3">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b4">f</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Olive" title="b3">N</font>,<font color="Olive" title="b2">G</font><br/>  for <font color="Olive" title="b5">g</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Olive" title="b1">H</font>,<font color="Olive" title="b2">G</font><br/>  for <font color="Olive" title="b6">phi</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Olive" title="b1">H</font>,<span class="p1">(<span class="default"><a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Olive" title="b3">N</font></span>)</span>  st (  for <font color="Olive" title="b7">n</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b3">N</font><br/>  for <font color="Olive" title="b8">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font> holds  <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b6">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">h</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b7">n</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b8">h</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span> ) holds <br/> ex <font color="Olive" title="b7">k</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_24.html#K1" title="GROUP_24:func.1">semidirect_product</a> (<font color="Olive" title="b3">N</font>,<font color="Olive" title="b1">H</font>,<font color="Olive" title="b6">phi</font>)</span>)</span>,<font color="Olive" title="b2">G</font> st <br/>( <font color="Olive" title="b4">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b7">k</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_24.html#K2" title="GROUP_24:func.2">incl1</a> (<font color="Olive" title="b3">N</font>,<font color="Olive" title="b1">H</font>,<font color="Olive" title="b6">phi</font>)</span>)</span> &amp; <font color="Olive" title="b5">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b7">k</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_24.html#K3" title="GROUP_24:func.3">incl2</a> (<font color="Olive" title="b3">N</font>,<font color="Olive" title="b1">H</font>,<font color="Olive" title="b6">phi</font>)</span>)</span> )</div></div>
