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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E12">IMDDX</font></span>: <a NAME="T10"><span class="comment"><font color="firebrick">:: DUALSP02:10</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>  st  not <font color="Olive" title="b1">X</font> is  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  holds <br/> ex <font color="Olive" title="b2">DuX</font> being    <a href="dualsp01.html#M1" title="DUALSP01:mode.1">SubRealNormSpace</a> of  <a href="dualsp01.html#K17" title="DUALSP01:func.17">DualSp</a> <span class="p1">(<span class="default"><a href="dualsp01.html#K17" title="DUALSP01:func.17">DualSp</a> <font color="Olive" title="b1">X</font></span>)</span> ex <font color="Olive" title="b3">L</font> being   <a href="lopban_1.html#V2" title="LOPBAN_1:attr.2">Lipschitzian</a>  <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">DuX</font> st <br/>( <font color="Olive" title="b3">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; <font color="Olive" title="b2">DuX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="normsp_3.html#K8" title="NORMSP_3:func.8">Im</a> <span class="p1">(<span class="default"><a href="dualsp02.html#K2" title="DUALSP02:func.2">BidualFunc</a> <font color="Olive" title="b1">X</font></span>)</span> &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">X</font> holds  <font color="Olive" title="b3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dualsp02.html#K1" title="DUALSP02:func.1">BiDual</a> <font color="Olive" title="b4">x</font> ) &amp; (  for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">X</font> holds  <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><font color="Olive" title="b4">x</font></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K2" title="NORMSP_0:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font></span>)</span></span><a href="normsp_0.html#K2" title="NORMSP_0:func.2">.||</a></span> ) )</div></div>
