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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E77">ThInjectiveIPO</font></span>: <a NAME="T92"><span class="comment"><font color="firebrick">:: GROUP_23:92</font></span><br/></a><div class="add"> 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">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">F</font> being   <a href="group_23.html#V5" title="GROUP_23:attr.5">normal</a>   <a href="group_20.html#M1" title="GROUP_20:mode.1">Subgroup-Family</a> of <font color="Olive" title="b2">I</font>,<font color="Olive" title="b1">G</font>  st <font color="Olive" title="b3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/>( <font color="Olive" title="b1">G</font> <a href="group_23.html#R1" title="GROUP_23:pred.1">is_internal_product_of</a> <font color="Olive" title="b3">F</font> iff (  <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> <span class="p1">(<span class="default"><a href="group_23.html#K3" title="GROUP_23:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <font color="Olive" title="b3">F</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">I</font><br/>  for <font color="Olive" title="b5">J</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b2">I</font>  st <font color="Olive" title="b5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> for <font color="Olive" title="b6">N</font> being   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <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>  st <font color="Olive" title="b6">N</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="group_23.html#K3" title="GROUP_23:func.3">Union</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K14" title="PRALG_1:func.14">Carrier</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">F</font> <a href="group_23.html#K19" title="GROUP_23:func.19">|</a> <font color="Olive" title="b5">J</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="group_23.html#K18" title="GROUP_23:func.18">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="group_6.html#K3" title="GROUP_6:func.3">/\</a> <font color="Olive" title="b6">N</font> <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> ) ) )</div></div>
