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<b>let </b><font color="Maroon" title="c1">n</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> ;<br/><b>let </b><font color="Maroon" title="c2">m</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">RAS</font> be    <a href="midsp_3.html#M2" title="MIDSP_3:mode.2">ReperAlgebra</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c4">W</font> be   <a href="midsp_2.html#NM1" title="MIDSP_2:NM.1">ATLAS</a> of <font color="Maroon" title="c3">RAS</font>;<br/>







<b>assume </b><a NAME="E1:55"/><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c3">RAS</font> <a href="midsp_3.html#R2" title="MIDSP_3:pred.2">is_semi_additive_in</a> <font color="Maroon" title="c2">m</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">p'm</font>, <font color="Maroon" title="c7">p''m</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>;<br/><b>let </b><font color="Maroon" title="c8">p</font> be   <a href="midsp_3.html#NM1" title="MIDSP_3:NM.1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">RAS</font>;<br/>







<b>assume </b><a NAME="E2:55"/>
<font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c7">p''m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c6">p'm</font>
 ;<br/>

<a NAME="E3:55"/><b>then </b>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c6">p'm</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c7">p''m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:55"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="midsp_3.html#T13" target="_self" title="MIDSP_3:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E4:55"/>
(  <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <font color="Maroon" title="c6">p'm</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">p</font></span>)</span> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p'm</font></span>)</span></span>)</span> iff <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c7">p''m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c6">p'm</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="midsp_2.html#T36" title="MIDSP_2:th.36">MIDSP_2:36</a></i>;<br/>


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