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<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">M2</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> )
 ;<br/>

<a NAME="E2:19"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1">MATRIXC1:1</a></i>;<br/>
<a NAME="E3:19"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1">MATRIXC1:1</a></i>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="matrixr1.html#K1">MXR2MXF</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrixr1.html#K5">-</a> <font color="Maroon">M2</font></span>)</span>;<br/>
<a NAME="E4:19"/><i><font color="Green">E44</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D15">VECTSP_1:def 15</a></i>;<br/>
<i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D15">VECTSP_1:def 15</a>, , <a class="ref" href="matrixr1.html#T1">MATRIXR1:1</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrixr1.html#K5">-</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_3.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p3">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrix_4.html#D1">MATRIX_4:def 1</a>, <a class="ref" href="vectsp_1.html#D15">VECTSP_1:def 15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K2">-</a> <span class="p3">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_3.html#D3">MATRIX_3:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M1</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr1.html#K1">MXR2MXF</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#D2">MATRIX_3:def 2</a>, </i>
;<br/>
<b>hence </b><a NAME="E7:19"/>
<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrixr1.html#K5">-</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i/>;<br/>


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