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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">M1</font>, <font color="Olive" title="b3">M2</font> being  <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Olive" title="b1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">M1</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  &amp; <font color="Olive" title="b3">M2</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b2">M1</font></span>)</span> <a href="matrix10.html#K3" title="MATRIX10:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b3">M2</font></span>)</span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">M1</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  &amp; <font color="Olive" title="b2">M2</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="matrix10.html#K3" title="MATRIX10:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Olive" title="b2">M2</font></span>)</span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">M1</font>, <font color="Maroon" title="c5">M2</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c3">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">M1</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  &amp; <font color="Maroon" title="c5">M2</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font></span>)</span> <a href="matrix10.html#K3" title="MATRIX10:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font></span>)</span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  )</span><br/>









<span class="kw">assume </span><a NAME="E1:58"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">M1</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  &amp; <font color="Maroon" title="c5">M2</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font></span>)</span> <a href="matrix10.html#K3" title="MATRIX10:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font></span>)</span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a> </span><br/>

<a NAME="E2:58"/><span class="kw">then </span>
( <font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix10.html#T42" target="_self" title="MATRIX10:th.42">Th42</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:58"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c4">M1</font></span>)</span> <a href="matrix10.html#K3" title="MATRIX10:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="matrix10.html#K5" title="MATRIX10:func.5">*</a> <font color="Maroon" title="c5">M2</font></span>)</span> is  <a href="matrix10.html#V4" title="MATRIX10:attr.4">Nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix10.html#T36" target="_self" title="MATRIX10:th.36">Th36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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