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<b>let </b><font color="Maroon">OAS</font> be   <a href="analoaf.html#NM1">OAffinSpace</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font>;<br/>









<b>assume </b><a NAME="E1:31"/><i><font color="Green">E29</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:31"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:31"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font> ) )
 
<b>by </b><i><a class="ref" href="diraf.html#T7">DIRAF:7</a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:31_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#T12">DIRAF:12</a></i>;<br/><b>hence </b><a NAME="E3:31_1"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T7">DIRAF:7</a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:31"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> ) )
 
<b>by </b><i><a class="ref" href="diraf.html#T4">DIRAF:4</a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E2:31_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_2_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/>
<a NAME="E2:31_2_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:31_2_1"/><b>then </b>
( <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:31_2_1"/>
<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/>


</div><b>end;</b></div><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_2"/><i><font color="Green">E37</font></i>: 
not  <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:31_2_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">c</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#T11">DIRAF:11</a>, <a class="ref" href="diraf.html#T37">DIRAF:37</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font><b> such that </b><br/><a NAME="E4:31_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E5:31_2_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E6:31_2_2"/>
( <font color="Maroon">c</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> )
 <b>by </b><i>, , <a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E7:31_2_2"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E8:31_2_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Olive">x</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_3"/><i><font color="Green">E40</font></i>: 
 <a href="diraf.html#R3">LIN</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_3_1"/><i><font color="Green">E42</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:31_2_3_1"/><i><font color="Green">E43</font></i>: 
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">c</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">c</font> ) )
 <b>by </b><i><a class="ref" href="diraf.html#T7">DIRAF:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_3_1_1"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 ;<br/><a NAME="E2:31_2_3_1_1"/><b>then </b>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">c</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> &amp;  <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E3:31_2_3_1_1"/><b>then </b>
( <font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E4:31_2_3_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T5">DIRAF:5</a></i>;<br/><a NAME="E5:31_2_3_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font> ) )
 <b>by </b><i>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/><a NAME="E6:31_2_3_1_1"/>
( <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> implies (  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">a</font> ) )
 <b>by </b><i><a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/><b>hence </b><a NAME="E7:31_2_3_1_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Olive">x</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 <b>by </b><i>, , <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:31_2_3_1"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Olive">x</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 <b>by </b><i>, , , <a class="ref" href="diraf.html#T21">DIRAF:21</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_3_2"/>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:31_2_3_2"/><b>then </b>
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">b</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13">DIRAF:13</a></i>;<br/><a NAME="E3:31_2_3_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E4:31_2_3_2"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><b>hence </b><a NAME="E5:31_2_3_2"/>
(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T14">DIRAF:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:31_2_3_3"/><i><font color="Green">E47</font></i>: 
 <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font>
 ;<br/><a NAME="E2:31_2_3_3"/><b>then </b>
( <font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#D3">DIRAF:def 3</a></i>;<br/><a NAME="E3:31_2_3_3"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="diraf.html#T6">DIRAF:6</a></i>;<br/><a NAME="E4:31_2_3_3"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="analoaf.html#D5">ANALOAF:def 5</a></i>;<br/><b>hence </b><a NAME="E5:31_2_3_3"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Olive">x</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:31_2_3"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Olive">x</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 <b>by </b><i>, , , <a class="ref" href="diraf.html#T35">DIRAF:35</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:31_2"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Olive">x</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:31"/>
 ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">OAS</font> st <br/>(  <a href="diraf.html#R1">Mid</a> <font color="Maroon">p</font>,<font color="Olive">x</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Olive">x</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
