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<span class="kw">let </span><font color="Maroon" title="c1">AFV</font> be   <a href="afvect0.html#NM1" title="AFVECT0:NM.1">WeakAffVect</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">q</font>, <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font> holds <br/> (  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Olive" title="b1">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Olive" title="b2">q</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">a</font></span>)</span> iff ( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Olive" title="b2">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AFV</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font></span>)</span> iff ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> ) )</span><br/>







<div><span class="lab"><font color="Green" title="E29">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_1"/>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> ) &amp; ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> ) )</span><br/><a NAME="E2:51_1"/><span class="kw">then </span>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T36" target="_self" title="AFVECT0:th.36">Th36</a></span>;<br/><a NAME="E3:51_1"/><span class="kw">then </span>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T44" target="_self" title="AFVECT0:th.44">Th44</a></span>;<br/><a NAME="E4:51_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T43" target="_self" title="AFVECT0:th.43">Th43</a></span>;<br/><a NAME="E5:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
(  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#D4" target="_self" title="AFVECT0:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E6:51_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T23" target="_self" title="AFVECT0:th.23">Th23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )</span><br/><span class="kw">thus </span><a NAME="E7:51_1"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="afvect0.html#T23" target="_self" title="AFVECT0:th.23">Th23</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="51_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:51_2"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font></span>)</span></span><br/><a NAME="E2:51_2"/><span class="kw">then </span>
(  <a href="afvect0.html#R2" title="AFVECT0:pred.2">Mid</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T23" target="_self" title="AFVECT0:th.23">Th23</a></span>;<br/><a NAME="E3:51_2"/><span class="kw">then </span>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#D4" target="_self" title="AFVECT0:def.4">Def4</a>, <a class="ref" href="afvect0.html#T43" target="_self" title="AFVECT0:th.43">Th43</a></span>;<br/><a NAME="E4:51_2"/><span class="kw">then </span>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<span class="p2">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T44" target="_self" title="AFVECT0:th.44">Th44</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_2"/>
 <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afvect0.html#T36" target="_self" title="AFVECT0:th.36">Th36</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="E3:51"/>
(  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c3">q</font>,<span class="p1">(<span class="default"><a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font></span>)</span> iff ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> or  <a href="afvect0.html#R1" title="AFVECT0:pred.1">MDist</a> <font color="Maroon" title="c3">q</font>, <a href="afvect0.html#K1" title="AFVECT0:func.1">PSym</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E29">A1</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>
