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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E19">Th19</font></span>: <a NAME="T19"><span class="comment"><font color="firebrick">:: MATRPROB:19</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">D</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">n</font>, <font color="Olive" title="b4">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">e1</font>, <font color="Olive" title="b6">e2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Olive" title="b1">D</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">m</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b6">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">m</font> holds <br/> ex <font color="Olive" title="b7">M</font> being   <a href="matrix_0.html#NM2" title="MATRIX_0:NM.2">Matrix</a> of <font color="Olive" title="b3">n</font>,<font color="Olive" title="b4">m</font>,<font color="Olive" title="b1">D</font> st <br/> for <font color="Olive" title="b8">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b8">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b3">n</font> holds <br/>( ( <font color="Olive" title="b8">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b2">k</font> implies <font color="Olive" title="b7">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e1</font> ) &amp; (  not <font color="Olive" title="b8">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Olive" title="b2">k</font> implies <font color="Olive" title="b7">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">e2</font> ) )</div></div>
