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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )</span><br/>



<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E32">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span><br/>

<a NAME="E2:71"/><b>then </b><i><font color="Green" title="E33">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T24" title="MATRIX_1:th.24">MATRIX_1:24</a></i>;<br/>
<a NAME="E3:71"/><b>then </b><i><font color="Green" title="E34">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E1:71"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="matrixr1.html#T61" target="_self" title="MATRIXR1:th.61">Th61</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K3" title="MATRIXR1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="txt" href="matrixr1.html#E1:71"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="matrixr1.html#E2:71"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="matrixr1.html#T36" target="_self" title="MATRIXR1:th.36">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixr1.html#E1:71"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="matrixr1.html#E2:71"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="matrixr1.html#T63" target="_self" title="MATRIXR1:th.63">Th63</a></i>
;<br/>
<a NAME="E5:71"/><b>then </b>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixr1.html#E2:71"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="matrixr1.html#E3:71"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="matrixr1.html#T3" target="_self" title="MATRIXR1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E6:71"/>
<span class="p1">(<span class="default"><a href="matrixr1.html#K8" title="MATRIXR1:func.8">0_Rmatrix</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrixr1.html#E3:71"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="matrixr1.html#T14" target="_self" title="MATRIXR1:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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