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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E12">Th9</font></span>: <a NAME="T9"><span class="comment"><font color="firebrick">:: RING_EMB:9</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a><br/>  for <font color="Olive" title="b3">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>  st <font color="Olive" title="b3">f</font> is  <a href="mod_4.html#V3" title="MOD_4:attr.3">monomorphism</a>  &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">f</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b4">C</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a> ex <font color="Olive" title="b5">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b6">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ex <font color="Olive" title="b7">G</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b4">C</font> st <br/>( <font color="Olive" title="b5">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b6">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">f</font></span>)</span> &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">X</font> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">X</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Olive" title="b1">A</font></span>)</span> &amp; <font color="Olive" title="b1">A</font> is    <a href="c0sp1.html#M1" title="C0SP1:mode.1">Subring</a> of <font color="Olive" title="b4">C</font> &amp; <font color="Olive" title="b7">G</font> is  <a href="quofield.html#NV5" title="QUOFIELD:NV.5">RingIsomorphism</a>  &amp;  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Olive" title="b1">A</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b7">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b3">f</font> )</div></div>
