<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="midsp_3.html#M2" target="_self">ReperAlgebra</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="midsp_2.html#NM1">ATLAS</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:57"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R1" target="_self">has_property_of_zero_in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>thus </b><a NAME="E2:57"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R3" target="_self">is_additive_in</a> <font color="Maroon">c<sub>2</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="midsp_2.html#NM2">Vector</a> of <font color="Maroon">c<sub>4</sub></font> holds  <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R3" target="_self">is_additive_in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:57_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R2" target="_self">is_semi_additive_in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57"><i><font color="Green">E38</font></i></a>, <a class="ref" href="midsp_3.html#T32" target="_self">Th32</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="midsp_2.html#NM2">Vector</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>



<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E4:57_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E5:57_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E7:57_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="midsp_1.html#D4">MIDSP_1:def 4</a></i>;<br/>
<i><font color="Green">E44</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="midsp_3.html#E7:57_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="midsp_2.html#T36">MIDSP_2:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E4:57_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="midsp_3.html#E5:57_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="midsp_3.html#D12" target="_self">Def12</a></i>
;<br/>
<a NAME="E9:57_1"/>
 <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="midsp_1.html#K4">@</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="midsp_3.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E10:57_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E2:57_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="midsp_3.html#E7:57_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="midsp_3.html#E53"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E11:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:57_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_2_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_2_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E8:57_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_1_2_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_2_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_1_2_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:57_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E45"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E13:57_1"/><i><font color="Green">E47</font></i>: 
 <a href="midsp_3.html#K9" target="_self">Phi</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E45"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E14:57_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:57_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_1_3_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:57_1_3_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_1_3_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:57_1_3_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E15:57_1"/>
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_3.html#E10:57_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="midsp_3.html#E12:57_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="midsp_3.html#E13:57_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="midsp_3.html#E45"><i><font color="Green">Lemma29</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:57"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="midsp_2.html#NM2">Vector</a> of <font color="Maroon">c<sub>4</sub></font> holds  <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> ) implies <font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R3" target="_self">is_additive_in</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="midsp_3.html#NM3">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="midsp_2.html#NM2">Vector</a> of <font color="Maroon">c<sub>4</sub></font> holds  <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:57_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R2" target="_self">is_semi_additive_in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E54"><i><font color="Green">Lemma37</font></i></a></i>;<br/>
<a NAME="E3:57_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="midsp_3.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font> st <font color="Olive">b<sub>4</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/> <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font></span>)</span> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>4</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="midsp_3.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>







<b>assume </b><a NAME="E1:57_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<a NAME="E2:57_2_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_3.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E3:57_2_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E5:57_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="midsp_1.html#D4">MIDSP_1:def 4</a></i>;<br/>
<i><font color="Green">E45</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="midsp_3.html#E5:57_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="midsp_2.html#T36">MIDSP_2:36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="midsp_3.html#D12" target="_self">Def12</a></i>
;<br/>
<a NAME="E7:57_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:57_2_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_2_1_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_2_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E6:57_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="midsp_2.html#T39">MIDSP_2:39</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_2_1_2_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_2_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E2:57_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_2_1_2_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:57_2_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E45"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E9:57_2_1"/><i><font color="Green">E47</font></i>: 
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E44"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E10:57_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:57_2_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be    <a href="midsp_3.html#M1" target="_self">Nat</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_2_1_3_1_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E3:57_2_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="midsp_3.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_2_1_3_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:57_2_1_3_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E2:57_2_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="midsp_2.html#K10">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_2_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="midsp_3.html#K5" target="_self">.</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="ref" href="midsp_3.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57_2_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="midsp_3.html#K6" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:57_2_1"/><b>then </b>
 <a href="midsp_3.html#K9" target="_self">Phi</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_3.html#K7" target="_self">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E45"><i><font color="Green">Lemma29</font></i></a></i>;<br/>
<a NAME="E12:57_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="midsp_2.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="midsp_3.html#E1:57_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="midsp_3.html#E8:57_2_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="midsp_3.html#E9:57_2_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:57_2_1"/>
 <a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="midsp_1.html#K4">@</a> <span class="p1">(<span class="default"><a href="midsp_3.html#K4" target="_self">*'</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="midsp_3.html#K1" target="_self">+*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="midsp_3.html#E2:57_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="midsp_3.html#E1:57_2_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="midsp_3.html#E5:57_2_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="midsp_3.html#E53"><i><font color="Green">Lemma36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:57_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="midsp_3.html#R3" target="_self">is_additive_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="midsp_3.html#D8" target="_self">Def8</a></i>;<br/>


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


</div>
