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<div class="add">

<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>thus </b><a NAME="E1:53"/>
( <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> implies for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">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="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">x</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E35">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">x</font> be   <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">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="c4">W</font>;<br/>

<b>consider </b><font color="Maroon" title="c6">a</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>;<br/>
<b>set </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<b>set </b><font color="Maroon" title="c9">p'</font> = <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font>;<br/>
<b>set </b><font color="Maroon" title="c10">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:53_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <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"><font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">i</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_1_1_1_1"/>
( <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><a NAME="E2:53_1_1_1_1_1"/><i><font color="Green" title="E37">A3</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_3.html#T15" target="_self" title="MIDSP_3:th.15">Th15</a></i>;<br/><a NAME="E3:53_1_1_1_1_1"/>
( <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="midsp_3.html#T17" target="_self" title="MIDSP_3:th.17">Th17</a></i>;<br/><a NAME="E4:53_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> &amp; <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> = 
<font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_3.html#E2:53_1_1_1_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <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"><font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font>
<b>by </b><i><a class="ref" href="midsp_3.html#T12" target="_self" title="MIDSP_3:th.12">Th12</a></i>
;<br/><b>hence </b><a NAME="E6:53_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <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"><font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E1:53_1_1_1_1_1"><i><font color="Green" title="E36">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_1_1_1_1_2"/><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> = 
<font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font></span>)</span> <a href="midsp_2.html#K10" title="MIDSP_2:func.10">.</a> <font color="Maroon" title="c4">W</font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:53_1_1_1_1_2"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self" title="MIDSP_3:def.11">Def11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <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"><font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:53_1_1_1_1_2"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <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"><font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c11">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:53_1"/><b>then </b>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T11" target="_self" title="MIDSP_3:th.11">Th11</a></i>;<br/>
<a NAME="E5:53_1"/><b>then </b>
 <a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:53_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="midsp_3.html#D7" target="_self" title="MIDSP_3:def.7">Def7</a></i>;<br/>
<a NAME="E6:53_1"/><b>then </b><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</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="c6">a</font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</a></i>;<br/>
<a NAME="E7:53_1"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W</font> <a href="midsp_2.html#K9" title="MIDSP_2:func.9">.</a> <font color="Maroon" title="c6">a</font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" title="MIDSP_3:func.4">*'</a> <font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" title="MIDSP_3:func.6">.</a> <font color="Maroon" title="c4">W</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E45"><i><font color="Green" title="E29">Lm5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:53_1"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_3.html#E6:53_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="midsp_3.html#E45"><i><font color="Green" title="E29">Lm5</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:53"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">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="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">x</font></span>)</span> ) implies <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_2"/><i><font color="Green" title="E35">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="midsp_3.html#NM3" title="MIDSP_3:NM.3">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="c4">W</font> holds   <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <font color="Olive" title="b1">x</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">a</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c3">RAS</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="midsp_3.html#D7" target="_self" title="MIDSP_3:def.7">MIDSP_3:def 7</a><br/><b>let </b><font color="Maroon" title="c6">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="c7">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="E2:53_2"/><i><font color="Green" title="E36">A7</font></i>: 
<font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'm</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">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"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">x'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:53_2"/>
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font> <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_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font>;<br/>
<a NAME="E1:53_2_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font> <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_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">i</font> be    <a href="midsp_3.html#M1" title="MIDSP_3:mode.1">Nat</a> of <font color="Maroon" title="c1">n</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:53_2_1_1_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> or <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:53_2_1_1_1_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><a NAME="E2:53_2_1_1_1_1_1"/><i><font color="Green" title="E38">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_3.html#T15" target="_self" title="MIDSP_3:th.15">Th15</a></i>;<br/><a NAME="E3:53_2_1_1_1_1_1"/><i><font color="Green" title="E39">A10</font></i>: 
( <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"><font color="Maroon" title="c7">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> &amp; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></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_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> )
 <b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></i>;<br/><a NAME="E4:53_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font> <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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_3.html#T12" target="_self" title="MIDSP_3:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E5:53_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> <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_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
 <b>by </b><i><a class="txt" href="midsp_3.html#E1:53_2_1_1_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="midsp_3.html#E2:53_2_1_1_1_1_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="midsp_3.html#E3:53_2_1_1_1_1_1"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="53_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:53_2_1_1_1_1_2"/><i><font color="Green" title="E37">A11</font></i>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">m</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></i>
<br/>.= 
<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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:53_2_1_1_1_1_2"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self" title="MIDSP_3:def.12">Def12</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font> <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_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c12">i</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:53_2_1"/>
<font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">p'</font> <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_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p1">(<span class="default"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_3.html#T16" target="_self" title="MIDSP_3:th.16">Th16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:53_2"/><b>then </b><i><font color="Green" title="E37">A12</font></i>: 
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <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"><a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p0">(<span class="default"><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"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>,<font color="Maroon" title="c7">p'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E44"><i><font color="Green" title="E28">Lm4</font></i></a></i>;<br/>
<a NAME="E5:53_2"/>
 <a href="midsp_3.html#K9" title="MIDSP_3:func.9">Phi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">W</font> <a href="midsp_3.html#K7" title="MIDSP_3:func.7">.</a> <font color="Maroon" title="c5">a</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E44"><i><font color="Green" title="E28">Lm4</font></i></a></i>;<br/>
<a NAME="E6:53_2"/><b>then </b>
<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>,<font color="Maroon" title="c7">p'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_2.html#K1" title="MIDSP_2:func.1">Double</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="c7">p'</font> <a href="midsp_3.html#K1" title="MIDSP_3:func.1">+*</a> <font color="Maroon" title="c2">m</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="midsp_1.html#K4" title="MIDSP_1:func.4">@</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">p'</font> <a href="midsp_3.html#K5" title="MIDSP_3:func.5">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:53_2"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="midsp_3.html#E4:53_2"><i><font color="Green" title="E37">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:53_2"/>
 <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="c7">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"><font color="Maroon" title="c5">a</font> <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>,<font color="Maroon" title="c7">p'</font></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_3.html#E2:53_2"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="midsp_2.html#T37" title="MIDSP_2:th.37">MIDSP_2:37</a></i>;<br/>


</div><b>end;</b></div>


</div>
