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<span class="kw">let </span><font color="Maroon" title="c1">AS</font> be   <a href="dirort.html#NM1" title="DIRORT:NM.1">Oriented_Orthogonality_Space</a>; <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="c1">AS</font> is  <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a>  &amp; ( <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V5" title="DIRORT:attr.5">Euclidean_like</a>  or <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a>  ) implies <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:33"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( not <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V5" title="DIRORT:attr.5">Euclidean_like</a>  &amp; not <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a>  ) or <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a>  )</span><br/>

<a NAME="E2:33"/>
( ( for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  holds <br/>not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> &amp; not <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> &amp;  ex <font color="Olive" title="b5">u</font>, <font color="Olive" title="b6">u1</font>, <font color="Olive" title="b7">v</font>, <font color="Olive" title="b8">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b5">u</font>,<font color="Olive" title="b6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">v</font>,<font color="Olive" title="b8">v1</font> &amp; not <font color="Olive" title="b7">v</font>,<font color="Olive" title="b8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b6">u1</font>,<font color="Olive" title="b5">u</font> ) ) ) implies for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">u2</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">w</font>, <font color="Olive" title="b8">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> &amp; <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> &amp; not <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u1</font> &amp; not <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> holds <br/><font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  holds <br/>not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> &amp; not <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> &amp;  ex <font color="Olive" title="b5">u</font>, <font color="Olive" title="b6">u1</font>, <font color="Olive" title="b7">v</font>, <font color="Olive" title="b8">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b5">u</font>,<font color="Olive" title="b6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">v</font>,<font color="Olive" title="b8">v1</font> &amp; not <font color="Olive" title="b7">v</font>,<font color="Olive" title="b8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b6">u1</font>,<font color="Olive" title="b5">u</font> ) )
 ; <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">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">u2</font>, <font color="Olive" title="b4">v</font>, <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">v2</font>, <font color="Olive" title="b7">w</font>, <font color="Olive" title="b8">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> &amp; <font color="Olive" title="b4">v</font>,<font color="Olive" title="b5">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font> &amp; <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> &amp; not <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u1</font> &amp; not <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">v1</font> holds <br/><font color="Olive" title="b3">u2</font>,<font color="Olive" title="b6">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b7">w</font>,<font color="Olive" title="b8">w1</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">u</font>, <font color="Maroon" title="c3">u1</font>, <font color="Maroon" title="c4">u2</font>, <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">v1</font>, <font color="Maroon" title="c7">v2</font>, <font color="Maroon" title="c8">w</font>, <font color="Maroon" title="c9">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</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">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> &amp; <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> &amp; <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> &amp; not <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> &amp; not <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> implies <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )</span><br/>















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:33_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font>
 <span class="kw">and </span><br/><a NAME="E3:33_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font>
 <span class="kw">and </span><br/><a NAME="E4:33_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</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">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E21">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> holds <br/><font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b2">u1</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"> ( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:33_1_1_1"/>
<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c6">v1</font>,<font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/><a NAME="E2:33_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c9">w1</font>,<font color="Maroon" title="c8">w</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dirort.html#D1" target="_self" title="DIRORT:def.1">Def1</a></span>;<br/><a NAME="E3:33_1_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
<font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c2">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:33_1_1_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">u1</font>
 <span class="kw">and </span><br/><a NAME="E5:33_1_1_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</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">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )</span><br/><a NAME="E6:33_1_1_1"/>
<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c2">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/><a NAME="E7:33_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">w1</font>,<font color="Maroon" title="c8">w</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:33_1_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E5:33_1_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E2:33_1_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E3:33_1_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="dirort.html#D5" target="_self" title="DIRORT:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E8:33_1_1_1"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_1"><span class="lab"><font color="Green" title="E22">A7</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="E3:33_1_1"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1_2"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">u1</font>, <font color="Olive" title="b3">v</font>, <font color="Olive" title="b4">v1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> holds <br/><font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Olive" title="b1">u</font>,<font color="Olive" title="b2">u1</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">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:33_1_2_1"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E1:33_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a></span>;<br/><a NAME="E2:33_1_2_1"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E1:33_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:33_1_2_1"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c2">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">u1</font>
 <span class="kw">and </span><br/><a NAME="E4:33_1_2_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">v1</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">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )</span><br/><a NAME="E5:33_1_2_1"/>
<font color="Maroon" title="c5">v</font>,<font color="Maroon" title="c6">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c3">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:33_1_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a></span>;<br/><a NAME="E6:33_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:33_1_2_1"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E4:33_1_2_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E2:33_1_2_1"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E1:33_1_2_1"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="dirort.html#D5" target="_self" title="DIRORT:def.5">Def5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:33_1_2_1"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1_2"><span class="lab"><font color="Green" title="E22">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><span class="kw">hence </span><a NAME="E3:33_1_2"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</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="E7:33_1"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> or <font color="Maroon" title="c4">u2</font>,<font color="Maroon" title="c7">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">w</font>,<font color="Maroon" title="c9">w1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E21">A6</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="E3:33"/>
( ( not <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V5" title="DIRORT:attr.5">Euclidean_like</a>  &amp; not <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a>  ) or <font color="Maroon" title="c1">AS</font> is  <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dirort.html#D6" target="_self" title="DIRORT:def.6">Def6</a>, <a class="ref" href="dirort.html#D7" target="_self" title="DIRORT:def.7">Def7</a>, <a class="ref" href="dirort.html#D8" target="_self" title="DIRORT:def.8">Def8</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>
