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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">AS</font> being  <a href="dirort.html#NM1" title="DIRORT:NM.1">Oriented_Orthogonality_Space</a>  st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> holds <br/>( <font color="Olive" title="b3">AS</font> is <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a> &amp; <font color="Olive" title="b3">AS</font> is <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a> &amp; <font color="Olive" title="b3">AS</font> is <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> &amp; <font color="Olive" title="b3">AS</font> is <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a> )</span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">AS</font> being  <a href="dirort.html#NM1" title="DIRORT:NM.1">Oriented_Orthogonality_Space</a>  st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Olive" title="b1">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> holds <br/>( <font color="Olive" title="b1">AS</font> is <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a> &amp; <font color="Olive" title="b1">AS</font> is <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a> &amp; <font color="Olive" title="b1">AS</font> is <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> &amp; <font color="Olive" title="b1">AS</font> is <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a> )</span><br/><b>let </b><font color="Maroon" title="c4">AS</font> be   <a href="dirort.html#NM1" title="DIRORT:NM.1">Oriented_Orthogonality_Space</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c4">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> implies ( <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a> ) )</span><br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E15">A1</font></i>: 
(  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c4">AS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a> )</span><br/>

<a NAME="E2:26"/><i><font color="Green" title="E16">A2</font></i>: 
 <a href="analort.html#K7" title="ANALORT:func.7">CMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="analort.html#K5" title="ANALORT:func.5">CORTM</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="analort.html#D8" title="ANALORT:def.8">ANALORT:def 8</a></i>;<br/>
<div><i><font color="Green" title="E17">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> )</span><br/><b>thus </b><a NAME="E1:26_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c13">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c14">v'</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c15">w'</font> = <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c16">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c17">v1'</font> = <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c18">w1'</font> = <font color="Maroon" title="c12">w1</font>, <font color="Maroon" title="c19">u2'</font> = <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c20">v2'</font> = <font color="Maroon" title="c10">v2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:26"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E2:26_1_1"/>
( <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c15">w'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w1'</font> &amp; not <font color="Maroon" title="c14">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v1'</font> implies <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T43" title="ANALORT:th.43">ANALORT:43</a></i>;<br/>
<b>hence </b><a NAME="E3:26_1_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> )
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E18">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> )</span><br/><b>thus </b><a NAME="E1:26_2"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c13">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c14">v'</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c15">w'</font> = <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c16">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c17">v1'</font> = <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c18">w1'</font> = <font color="Maroon" title="c12">w1</font>, <font color="Maroon" title="c19">u2'</font> = <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c20">v2'</font> = <font color="Maroon" title="c10">v2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:26"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E2:26_2_1"/>
( <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">v1'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c15">w'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w1'</font> &amp; not <font color="Maroon" title="c14">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v1'</font> implies <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T47" title="ANALORT:th.47">ANALORT:47</a></i>;<br/>
<b>hence </b><a NAME="E3:26_2_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c11">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> )
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E19">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">w1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> )</span><br/><b>thus </b><a NAME="E1:26_3"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c13">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c14">v'</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c15">w'</font> = <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c16">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c17">v1'</font> = <font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c18">w1'</font> = <font color="Maroon" title="c12">w1</font>, <font color="Maroon" title="c19">u2'</font> = <font color="Maroon" title="c7">u2</font>, <font color="Maroon" title="c20">v2'</font> = <font color="Maroon" title="c10">v2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:26"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E2:26_3_1"/>
( <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c14">v'</font>,<font color="Maroon" title="c17">v1'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c13">u'</font>,<font color="Maroon" title="c16">u1'</font>,<font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> &amp; not <font color="Maroon" title="c13">u'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">u1'</font> &amp; not <font color="Maroon" title="c14">v'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">v1'</font> implies <font color="Maroon" title="c19">u2'</font>,<font color="Maroon" title="c20">v2'</font>,<font color="Maroon" title="c15">w'</font>,<font color="Maroon" title="c18">w1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T49" title="ANALORT:th.49">ANALORT:49</a></i>;<br/>
<b>hence </b><a NAME="E3:26_3_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c8">v</font>,<font color="Maroon" title="c9">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> &amp; <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> &amp; not <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u1</font> &amp; not <font color="Maroon" title="c8">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> implies <font color="Maroon" title="c7">u2</font>,<font color="Maroon" title="c10">v2</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c12">w1</font> )
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c8">v1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> implies <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> )</span><br/><b>thus </b><a NAME="E1:26_4"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> implies <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c9">u'</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c10">v'</font> = <font color="Maroon" title="c7">v</font>, <font color="Maroon" title="c11">u1'</font> = <font color="Maroon" title="c6">u1</font>, <font color="Maroon" title="c12">v1'</font> = <font color="Maroon" title="c8">v1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E2:26"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E2:26_4_1"/>
( <font color="Maroon" title="c9">u'</font>,<font color="Maroon" title="c11">u1'</font>,<font color="Maroon" title="c10">v'</font>,<font color="Maroon" title="c12">v1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c10">v'</font>,<font color="Maroon" title="c12">v1'</font>,<font color="Maroon" title="c9">u'</font>,<font color="Maroon" title="c11">u1'</font> <a href="analort.html#R2" title="ANALORT:pred.2">are_COrtm_wrt</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T19" title="ANALORT:th.19">ANALORT:19</a></i>;<br/>
<b>hence </b><a NAME="E3:26_4_1"/>
( <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> implies <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c8">v1</font> <a href="dirort.html#NR1" title="DIRORT:NR.1">'//'</a> <font color="Maroon" title="c5">u</font>,<font color="Maroon" title="c6">u1</font> )
 <b>by </b><i><a class="txt" href="#E1:26"><i><font color="Green" title="E15">A1</font></i></a>, <a class="ref" href="analort.html#T57" title="ANALORT:th.57">ANALORT:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:26"/>
( <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V6" title="DIRORT:attr.6">Minkowskian_like</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V4" title="DIRORT:attr.4">left_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V3" title="DIRORT:attr.3">right_transitive</a> &amp; <font color="Maroon" title="c4">AS</font> is <a href="dirort.html#V2" title="DIRORT:attr.2">bach_transitive</a> )
 <b>by </b><i><a class="txt" href="#E3:26"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="#E4:26"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="#E5:26"><i><font color="Green" title="E19">A5</font></i></a>, <a class="ref" href="dirort.html#D4" target="_self" title="DIRORT:def.4">Def4</a>, <a class="ref" href="dirort.html#D5" target="_self" title="DIRORT:def.5">Def5</a>, <a class="ref" href="dirort.html#D6" target="_self" title="DIRORT:def.6">Def6</a>, <a class="ref" href="dirort.html#D8" target="_self" title="DIRORT:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
