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<div><span class="kw">theorem </span><a NAME="T48"><span class="comment"><font color="firebrick">:: ZF_LANG:48</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Olive" title="b2">H</font> is  <a href="zf_lang.html#V10" title="ZF_LANG:attr.10">biconditional</a>  holds <br/>( ( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K26" title="ZF_LANG:func.26">the_left_side_of</a> <font color="Olive" title="b2">H</font> implies  ex <font color="Olive" title="b3">G</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> st <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <font color="Olive" title="b3">G</font> ) &amp; (  ex <font color="Olive" title="b3">G</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> st <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <font color="Olive" title="b3">G</font> implies <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K26" title="ZF_LANG:func.26">the_left_side_of</a> <font color="Olive" title="b2">H</font> ) &amp; ( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K27" title="ZF_LANG:func.27">the_right_side_of</a> <font color="Olive" title="b2">H</font> implies  ex <font color="Olive" title="b3">G</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> st <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <font color="Olive" title="b1">F</font> ) &amp; (  ex <font color="Olive" title="b3">G</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> st <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">G</font> <a href="zf_lang.html#K11" title="ZF_LANG:func.11">&lt;=&gt;</a> <font color="Olive" title="b1">F</font> implies <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K27" title="ZF_LANG:func.27">the_right_side_of</a> <font color="Olive" title="b2">H</font> ) ) <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#D37" target="_self" onmouseover="rs('zf_lang/D37')" onmouseout="rh()">Def37</a>, <a class="ref" href="zf_lang.html#D38" target="_self" onmouseover="rs('zf_lang/D38')" onmouseout="rh()">Def38</a></span>;<br/></div></div>
