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<div><span class="kw">theorem </span><a NAME="T47"><span class="comment"><font color="firebrick">:: GROUP_1A:47</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="group_1a.html#V1" title="GROUP_1A:attr.1">add-unital</a>   <a href="algstr_0.html#L1" title="ALGSTR_0:struct.1">addMagma</a> <br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b4">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <span class="p1">(<span class="default"><a href="group_1a.html#K4" title="GROUP_1A:func.4">mult</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Olive" title="b4">n</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">y</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"><a href="group_1a.html#K4" title="GROUP_1A:func.4">mult</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Olive" title="b4">n</font>,<font color="Olive" title="b2">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1a.html#K4" title="GROUP_1A:func.4">mult</a> <font color="Olive" title="b1">L</font></span>)</span> <a href="nat_1.html#K10" title="NAT_1:func.10">.</a> (<font color="Olive" title="b4">n</font>,<font color="Olive" title="b3">y</font>)</span>)</span></div></div>
