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<div>:: <span class="kw">deftheorem </span><span class="lab"><font color="Green" title="E25">Def1</font></span>   defines <a href="group_24.html#K1" title="GROUP_24:func.1">semidirect_product</a> <a onclick="hs(this)" href="javascript:()">GROUP_24:def 1 : <br/></a><span> for <font color="Olive" title="b1">G</font>, <font color="Olive" title="b2">A</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b3">phi</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <font color="Olive" title="b2">A</font>,<span class="p1">(<span class="default"><a href="autgroup.html#K3" title="AUTGROUP:func.3">AutGroup</a> <font color="Olive" title="b1">G</font></span>)</span><br/>  for <font color="Olive">b<sub>4</sub></font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  holds <br/> ( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_24.html#K1" title="GROUP_24:func.1">semidirect_product</a> (<font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">phi</font>) iff (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">A</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; (  for <font color="Olive" title="b5">f</font>, <font color="Olive" title="b6">g</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">G</font>,<font color="Olive" title="b2">A</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>  ex <font color="Olive" title="b7">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ex <font color="Olive" title="b8">a1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">A</font> ex <font color="Olive" title="b9">g2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive">b<sub>4</sub></font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">f</font>,<font color="Olive" title="b6">g</font>) &amp; <font color="Olive" title="b8">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 &amp; <font color="Olive" title="b9">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">phi</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b8">a1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">g2</font></span>)</span>) &amp; <font color="Olive" title="b7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b2">A</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Olive" title="b5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>) ) ) ) );<br/></span></div>
