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<div><span class="kw">theorem </span><a NAME="T33"><span class="comment"><font color="firebrick">:: RINGFRAC:33</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font> being   non  <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a><br/>  for <font color="Olive" title="b2">S</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a>   <a href="ringfrac.html#V2" title="RINGFRAC:attr.2">without_zero</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">A</font>  st  not  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">S</font> &amp; <font color="Olive" title="b1">A</font> is   <a href="vectsp_2.html#NM2" title="VECTSP_2:NM.2">domRing</a> holds <br/>(  <a href="vectsp10.html#K8" title="VECTSP10:func.8">ker</a> <span class="p1">(<span class="default"><a href="ringfrac.html#K16" title="RINGFRAC:func.16">canHom</a> <font color="Olive" title="b2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">A</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  <a href="ringfrac.html#K16" title="RINGFRAC:func.16">canHom</a> <font color="Olive" title="b2">S</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</div></div>
