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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">t</font>, <font color="Olive" title="b4">y</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">z</font>,<font color="Olive" title="b2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">t</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">z</font> holds <br/> ex <font color="Olive" title="b5">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b4">y</font>,<font color="Olive" title="b2">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b5">u</font> &amp; <font color="Olive" title="b4">y</font>,<font color="Olive" title="b1">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b5">u</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">z</font>, <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">t</font>, <font color="Maroon" title="c5">y</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">t</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">z</font> implies  ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">t</font>
 <span class="kw">and </span><br/><a NAME="E2:39"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">z</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">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E26">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E2:39_1"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">and </span><br/><a NAME="E3:39_1"/><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T17" target="_self" title="DIRAF:th.17">Th17</a></span>;<br/><a NAME="E4:39_1"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="diraf.html#D3" target="_self" title="DIRAF:def.3">Def3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E6:39_1"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">q</font>
 <span class="kw">and </span><br/><a NAME="E7:39_1"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</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="E8:39_1"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c3">x</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">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )</span><br/><a NAME="E9:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E10:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T5" target="_self" title="DIRAF:th.5">Th5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E12:39_1"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">r</font>
 <span class="kw">and </span><br/><a NAME="E13:39_1"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
<font color="Maroon" title="c7">q</font>,<font color="Maroon" title="c6">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c8">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E28">A5</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="E35">A12</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</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">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )</span><br/><a NAME="E2:39_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A13</font></span>: 
<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="ref" href="diraf.html#T5" target="_self" title="DIRAF:th.5">Th5</a></span>;<br/><a NAME="E3:39_1_1"/>
<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="diraf.html#T5" target="_self" title="DIRAF:th.5">Th5</a></span>;<br/><a NAME="E4:39_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E5:39_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E6:39_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">t</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>;<br/><a NAME="E7:39_1_1"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c4">t</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="E8:39_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c4">t</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>;<br/><span class="kw">hence </span><a NAME="E9:39_1_1"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1"><span class="lab"><font color="Green" title="E37">A14</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><div><span class="lab"><font color="Green" title="E36">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:39_1_2"/><span class="lab"><font color="Green" title="E37">A16</font></span>: 
<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="ref" href="diraf.html#T5" target="_self" title="DIRAF:th.5">Th5</a></span>;<br/><span class="kw">assume </span><a NAME="E2:39_1_2"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</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">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )</span><br/><a NAME="E3:39_1_2"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="diraf.html#T5" target="_self" title="DIRAF:th.5">Th5</a></span>;<br/><a NAME="E4:39_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c2">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E2:39_1_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a>, <a class="ref" href="diraf.html#T6" target="_self" title="DIRAF:th.6">Th6</a></span>;<br/><a NAME="E5:39_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="txt" href="#E1:39_1_2"><span class="lab"><font color="Green" title="E37">A16</font></span></a>, <a class="ref" href="diraf.html#T6" target="_self" title="DIRAF:th.6">Th6</a></span>;<br/><a NAME="E6:39_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c3">x</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>;<br/><a NAME="E7:39_1_2"/>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T7" target="_self" title="DIRAF:th.7">Th7</a></span>;<br/><a NAME="E8:39_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c3">x</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>;<br/><span class="kw">hence </span><a NAME="E9:39_1_2"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_2"><span class="lab"><font color="Green" title="E39">A18</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:39_1_3"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">and </span><br/><a NAME="E2:39_1_3"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</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">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )</span><br/><a NAME="E3:39_1_3"/>
<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="diraf.html#T5" target="_self" title="DIRAF:th.5">Th5</a></span>;<br/><a NAME="E4:39_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E2:39_1_3"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="ref" href="diraf.html#T6" target="_self" title="DIRAF:th.6">Th6</a></span>;<br/><a NAME="E5:39_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A21</font></span>: 
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c8">r</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>;<br/><a NAME="E6:39_1_3"/>
<font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="ref" href="diraf.html#T5" target="_self" title="DIRAF:th.5">Th5</a></span>;<br/><a NAME="E7:39_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">r</font>,<font color="Maroon" title="c4">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="txt" href="#E1:39_1_3"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="diraf.html#T6" target="_self" title="DIRAF:th.6">Th6</a></span>;<br/><a NAME="E8:39_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c8">r</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>;<br/><span class="kw">hence </span><a NAME="E9:39_1_3"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_3"><span class="lab"><font color="Green" title="E39">A21</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">hence </span><a NAME="E17:39_1"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:39_1"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="txt" href="#E15:39_1"><span class="lab"><font color="Green" title="E36">A15</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:39_2"/>
<font color="Maroon" title="c2">z</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">t</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">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><br/><a NAME="E3:39_2"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c6">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E4:39_2"/>
( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c6">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c6">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="ref" href="diraf.html#D4" target="_self" title="DIRAF:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:39_2"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )
 ; <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="E5:39"/>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c3">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">u</font> &amp; <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">z</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">t</font>,<font color="Olive" title="b1">u</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E26">A3</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
