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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E5">Th12</font></span>: <a NAME="T10"><span class="comment"><font color="firebrick">:: LOPBAN15:10</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">X</font> being   <a href="prvect_2.html#NM2" title="PRVECT_2:NM.2">RealLinearSpace-Sequence</a><br/>  for <font color="Olive" title="b2">r</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <br/>  for <font color="Olive" title="b3">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Olive" title="b1">X</font></span>)</span><br/>  for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="prvect_1.html#K15" title="PRVECT_1:func.15">carr</a> <font color="Olive" title="b1">X</font></span>)</span> holds <br/> ( <span class="p1">(<span class="default"><span class="p2"><a href="prvect_2.html#K2" title="PRVECT_2:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K5" title="PRVECT_2:func.5">multop</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="prvect_2.html#K2" title="PRVECT_2:func.2">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">r</font>,<font color="Olive" title="b3">v</font>)</span>)</span> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="prvect_2.html#K4" title="PRVECT_2:func.4">.</a> <font color="Olive" title="b4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">r</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">.</a> <font color="Olive" title="b4">i</font></span>)</span>) &amp; (  for <font color="Olive" title="b5">vi</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="prvect_2.html#K4" title="PRVECT_2:func.4">.</a> <font color="Olive" title="b4">i</font></span>)</span>  st <font color="Olive" title="b5">vi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">.</a> <font color="Olive" title="b4">i</font> holds <br/><span class="p1">(<span class="default"><span class="p2"><a href="prvect_2.html#K2" title="PRVECT_2:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="prvect_2.html#K5" title="PRVECT_2:func.5">multop</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="prvect_2.html#K2" title="PRVECT_2:func.2">:]</a></span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Olive" title="b2">r</font>,<font color="Olive" title="b3">v</font>)</span>)</span> <a href="prvect_1.html#K8" title="PRVECT_1:func.8">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b5">vi</font> ) )</div></div>
