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<div><span class="kw">theorem </span><a NAME="T41"><span class="comment"><font color="firebrick">:: RINGFRAC:41</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="b3">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><br/>  for <font color="Olive" title="b4">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="b4">f</font> is  <a href="quofield.html#NV1" title="QUOFIELD:NV.1">RingHomomorphism</a>  &amp; <font color="Olive" title="b4">f</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">.:</a> <font color="Olive" title="b3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ringfrac.html#K3" title="RINGFRAC:func.3">Unit_Set</a> <font color="Olive" title="b2">B</font> holds <br/> <a href="ringfrac.html#K20" title="RINGFRAC:func.20">Univ_Map</a> (<font color="Olive" title="b3">S</font>,<font color="Olive" title="b4">f</font>) is  <a href="quofield.html#NV1" title="QUOFIELD:NV.1">RingHomomorphism</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ringfrac.html#T38" target="_self" onmouseover="rs('ringfrac/T38')" onmouseout="rh()">Th57</a>, <a class="ref" href="ringfrac.html#T39" target="_self" onmouseover="rs('ringfrac/T39')" onmouseout="rh()">Th58</a>, <a class="ref" href="ringfrac.html#T40" target="_self" onmouseover="rs('ringfrac/T40')" onmouseout="rh()">Th59</a></span>;<br/></div></div>
