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<span class="kw">let </span><font color="Maroon" title="c1">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>  st  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">b</font> holds <br/> ex <font color="Olive" title="b5">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">x</font>,<font color="Olive" title="b4">a</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">c</font>,<font color="Olive" title="b5">x</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font> implies  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> ) )</span><br/>









<a NAME="E1:32"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T4" title="DIRAF:th.4">DIRAF:4</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:32"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E16">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> )</span><br/><a NAME="E2:32_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:32_1"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:32"/><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E18">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:32_2"/><span class="lab"><font color="Green" title="E19">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E2:32_2"/><span class="lab"><font color="Green" title="E20">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E3:32_2"/><span class="lab"><font color="Green" title="E21">A8</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )</span><br/><a NAME="E4:32_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E5:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A9</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E6:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A10</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E7:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A11</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_2"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E5:32_2"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><div><span class="lab"><font color="Green" title="E25">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E26">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_2_1_1"/><span class="lab"><font color="Green" title="E27">A14</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )</span><br/><div><span class="lab"><font color="Green" title="E28">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:32_2_1_1_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E2:32_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E3:32_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A16</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/><span class="kw">assume </span><a NAME="E4:32_2_1_1_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )</span><br/><a NAME="E5:32_2_1_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E6:32_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E7:32_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/><a NAME="E8:32_2_1_1_1"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
( <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/><a NAME="E9:32_2_1_1_1"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_2"><span class="lab"><font color="Green" title="E21">A8</font></span></a>, <a class="txt" href="#E5:32_2"><span class="lab"><font color="Green" title="E22">A9</font></span></a>, <a class="txt" href="#E6:32_2"><span class="lab"><font color="Green" title="E23">A10</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/><span class="kw">hence </span><a NAME="E10:32_2_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_2_1_1_1"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="txt" href="#E3:32_2_1_1_1"><span class="lab"><font color="Green" title="E29">A16</font></span></a>, <a class="txt" href="#E8:32_2_1_1_1"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:32_2_1_1"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c3">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:32_2_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:32_2_1_1"><span class="lab"><font color="Green" title="E27">A14</font></span></a>, <a class="txt" href="#E2:32_2_1_1"><span class="lab"><font color="Green" title="E28">A15</font></span></a>, <a class="ref" href="diraf.html#T21" title="DIRAF:th.21">DIRAF:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E27">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_2_1_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> )</span><br/><a NAME="E2:32_2_1_2"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E3:32_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E4:32_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:32_2_1_2"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_2"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E7:32_2"><span class="lab"><font color="Green" title="E24">A11</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><span class="lab"><font color="Green" title="E28">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:32_2_1_3"/><span class="lab"><font color="Green" title="E29">A21</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E2:32_2_1_3"/><span class="lab"><font color="Green" title="E30">A22</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )</span><br/><a NAME="E3:32_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E4:32_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_2"><span class="lab"><font color="Green" title="E20">A7</font></span></a>, <a class="txt" href="#E1:32_2_1_3"><span class="lab"><font color="Green" title="E29">A21</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E5:32_2_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E6:32_2_1_3"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_2_1_3"><span class="lab"><font color="Green" title="E30">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E4:32_2_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )</span><br/><span class="kw">hence </span><a NAME="E5:32_2_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2_1"><span class="lab"><font color="Green" title="E26">A13</font></span></a>, <a class="txt" href="#E2:32_2_1"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E3:32_2_1"><span class="lab"><font color="Green" title="E28">A20</font></span></a>, <a class="ref" href="diraf.html#T35" title="DIRAF:th.35">DIRAF:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:32_2_2"/><span class="lab"><font color="Green" title="E26">A23</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></span>;<br/><span class="kw">assume </span><a NAME="E2:32_2_2"/><span class="lab"><font color="Green" title="E27">A24</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )</span><br/><a NAME="E3:32_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font><span class="kw"> such that </span><br/><a NAME="E5:32_2_2"/><span class="lab"><font color="Green" title="E28">A25</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E6:32_2_2"/><span class="lab"><font color="Green" title="E29">A26</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2_2"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="ref" href="pasch.html#T25" target="_self" title="PASCH:th.25">Th25</a></span>;<br/><a NAME="E7:32_2_2"/><span class="lab"><font color="Green" title="E30">A27</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_2_2"><span class="lab"><font color="Green" title="E28">A25</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/><a NAME="E8:32_2_2"/>
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_2_2"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:32_2_2"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E2:32_2_2"><span class="lab"><font color="Green" title="E27">A24</font></span></a>, <a class="txt" href="#E6:32_2_2"><span class="lab"><font color="Green" title="E29">A26</font></span></a>, <a class="txt" href="#E7:32_2_2"><span class="lab"><font color="Green" title="E30">A27</font></span></a>, <a class="ref" href="pasch.html#T22" target="_self" title="PASCH:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:32_2"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E25">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:32"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:32"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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