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<div>:: <span class="kw">deftheorem </span>   defines <a href="group_23.html#R2" title="GROUP_23:pred.2">is_internal_product_of</a> <a onclick="hs(this)" href="javascript:()">GROUP_23:def 24 : <br/></a><span> for <font color="Olive" title="b1">G</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><br/>  for <font color="Olive" title="b2">Fam</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="group_3.html#K1" title="GROUP_3:func.1">Subgroups</a> <font color="Olive" title="b1">G</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">G</font> <a href="group_23.html#R2" title="GROUP_23:pred.2">is_internal_product_of</a> <font color="Olive" title="b2">Fam</font> iff ( (  for <font color="Olive" title="b3">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b3">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">Fam</font> holds <br/><font color="Olive" title="b3">H</font> is   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font> ) &amp;  ex <font color="Olive" title="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b4">UH</font> where <font color="Olive" title="b4">UH</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">G</font> :  ex <font color="Olive" title="b5">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b5">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">Fam</font> &amp; <font color="Olive" title="b4">UH</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b5">H</font> ) </span> } </span>  &amp;  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font>, the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Olive" title="b3">A</font> ) &amp; (  for <font color="Olive" title="b3">H</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b3">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">Fam</font> holds <br/> for <font color="Olive" title="b4">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">G</font>  st <font color="Olive" title="b4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b5">UK</font> where <font color="Olive" title="b5">UK</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">G</font> :  ex <font color="Olive" title="b6">K</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b6">K</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">Fam</font> &amp; <font color="Olive" title="b5">UK</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b6">K</font> &amp; <font color="Olive" title="b6">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">H</font> ) </span> } </span>  holds <br/><font color="Olive" title="b3">H</font> <a href="group_2.html#K10" title="GROUP_2:func.10">/\</a> <span class="p1">(<span class="default"><a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <font color="Olive" title="b4">A</font></span>)</span> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K6" title="GROUP_2:func.6">(1).</a> <font color="Olive" title="b1">G</font> ) ) );<br/></span></div>
