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<span class="kw">let </span><font color="Maroon" title="c1">S</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font> holds <br/><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )</span><br/>







<div><span class="lab"><font color="Green" title="E18">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_1"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )</span><br/><a NAME="E2:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</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="E3:28_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</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="E4:28_1"/>
( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" target="_self" title="DIRAF:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E19">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E20">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_2_1"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )</span><br/><a NAME="E2:28_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T2" target="_self" title="DIRAF:th.2">Th5</a></span>;<br/><a NAME="E3:28_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</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="E4:28_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T2" target="_self" title="DIRAF:th.2">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:28_2_1"/>
( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" target="_self" title="DIRAF:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E21">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_2_2"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )</span><br/><a NAME="E2:28_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c3">y</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="E3:28_2_2"/>
( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" target="_self" title="DIRAF:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E3:28_2"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )</span><br/><span class="kw">hence </span><a NAME="E4:28_2"/>
( <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font> implies <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E1:28_2"><span class="lab"><font color="Green" title="E20">A3</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E3:28"/>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font></span><br/>

<span class="kw">hence </span><a NAME="E4:28"/>
<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="diraf.html#D4" target="_self" title="DIRAF:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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