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<div><span class="kw">theorem </span><a NAME="T10"><span class="comment"><font color="firebrick">:: LOPBAN12:10</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">Z</font> being   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>  ex <font color="Olive" title="b4">I</font> being   <a href="lopban_1.html#NM1" title="LOPBAN_1:NM.1">LinearOperator</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" title="LOPBAN_1:func.8">R_VectorSpace_of_LinearOperators</a> (<font color="Olive" title="b1">X</font>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8" title="LOPBAN_1:func.8">R_VectorSpace_of_LinearOperators</a> (<font color="Olive" title="b2">Y</font>,<font color="Olive" title="b3">Z</font>)</span>)</span>)</span>)</span>,<span class="p1">(<span class="default"><a href="lopban10.html#K4" title="LOPBAN10:func.4">R_VectorSpace_of_MultilinearOperators</a> (<span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">X</font>,<font color="Olive" title="b2">Y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<font color="Olive" title="b3">Z</font>)</span>)</span> st <br/>( <font color="Olive" title="b4">I</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b5">u</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="lopban_1.html#K8" title="LOPBAN_1:func.8">R_VectorSpace_of_LinearOperators</a> (<font color="Olive" title="b1">X</font>,<span class="p2">(<span class="default"><a href="lopban_1.html#K8" title="LOPBAN_1:func.8">R_VectorSpace_of_LinearOperators</a> (<font color="Olive" title="b2">Y</font>,<font color="Olive" title="b3">Z</font>)</span>)</span>)</span>)</span><br/>  for <font color="Olive" title="b6">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b1">X</font><br/>  for <font color="Olive" title="b7">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Olive" title="b2">Y</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b5">u</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b6">x</font>,<font color="Olive" title="b7">y</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">u</font> <a href="lopban_1.html#K9" title="LOPBAN_1:func.9">.</a> <font color="Olive" title="b6">x</font></span>)</span> <a href="lopban_1.html#K9" title="LOPBAN_1:func.9">.</a> <font color="Olive" title="b7">y</font> ) )</div></div>
