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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">M2</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:59"/>
<font color="Maroon" title="c2">M1</font> <a href="matrix10.html#R1" title="MATRIX10:pred.1">is_less_than</a> <font color="Maroon" title="c3">M2</font>
 ;<br/>

<a NAME="E2:59"/><b>then </b>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M1</font> holds <br/><font color="Maroon" title="c2">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<b>by </b><i><a class="ref" href="matrix10.html#D5" target="_self" title="MATRIX10:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E3:59"/>
<font color="Maroon" title="c2">M1</font> <a href="matrix10.html#R2" title="MATRIX10:pred.2">is_less_or_equal_with</a> <font color="Maroon" title="c3">M2</font>
 <b>by </b><i><a class="ref" href="matrix10.html#D6" target="_self" title="MATRIX10:def.6">Def6</a></i>;<br/>


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