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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">w</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">p</font>, <font color="Olive" title="b4">q</font>, <font color="Olive" title="b5">p1</font>, <font color="Olive" title="b6">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font></span>)</span>  st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b3">p</font>,<font color="Olive" title="b4">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b5">p1</font>,<font color="Olive" title="b6">p2</font> &amp; <font color="Olive" title="b5">p1</font>,<font color="Olive" title="b4">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b6">p2</font>,<font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b6">p2</font>,<font color="Olive" title="b4">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">p</font>,<font color="Olive" title="b5">p1</font></span><br/><b>let </b><font color="Maroon" title="c2">w</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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span>  st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font> &amp; <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b2">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b4">p2</font>,<font color="Olive" title="b1">p</font> holds <br/><font color="Olive" title="b4">p2</font>,<font color="Olive" title="b2">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">p1</font></span><br/><b>let </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="analmetr.html#K2" title="ANALMETR:func.2">AMSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span>; <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">w</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> &amp; <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c5">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c5">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">p1</font> )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:46"/><i><font color="Green" title="E38">A1</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 <b>and </b><br/><a NAME="E2:46"/><i><font color="Green" title="E39">X</font></i>: 
<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font>
 <b>and </b><br/><a NAME="E3:46"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c5">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c5">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">p1</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c9">v</font> = <font color="Maroon" title="c5">q</font>, <font color="Maroon" title="c10">u1</font> = <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c11">u2</font> = <font color="Maroon" title="c7">p2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E5:46"/>
( <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">u1</font>,<font color="Maroon" title="c11">u2</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c10">u1</font>,<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c11">u2</font>,<font color="Maroon" title="c8">u</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="analmetr.html#E3:46"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="analmetr.html#T31" target="_self" title="ANALMETR:th.31">Th31</a>, <a class="txt" href="analmetr.html#E2:46"><i><font color="Green" title="E39">X</font></i></a></i>;<br/>
<a NAME="E6:46"/><b>then </b><i><font color="Green" title="E41">A3</font></i>: 
( <font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c11">u2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">u1</font> <a href="analmetr.html#R2" title="ANALMETR:pred.2">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">u1</font>,<font color="Maroon" title="c8">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">u2</font> <a href="analmetr.html#R2" title="ANALMETR:pred.2">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="analmetr.html#D3" target="_self" title="ANALMETR:def.3">Def3</a></i>;<br/>
<div><i><font color="Green" title="E42">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">u</font>, <font color="Maroon" title="c13">v</font>, <font color="Maroon" title="c14">w</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">v</font></span><br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c14">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c12">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47" title="RLVECT_1:th.47">RLVECT_1:47</a></i>
<br/>.= 
<font color="Maroon" title="c14">w</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c13">v</font>
<b>by </b><i><a class="ref" href="analoaf.html#T4" title="ANALOAF:th.4">ANALOAF:4</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>
<a NAME="E8:46"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">u1</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">u2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">u1</font>
 
;<br/>
<a NAME="E9:46"/><i><font color="Green" title="E44">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">u2</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">u2</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E7:46"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<a NAME="E10:46"/><i><font color="Green" title="E45">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">u1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">u1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c8">u</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E7:46"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<a NAME="E11:46"/>
<font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c11">u2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c8">u</font></span>)</span> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c10">u1</font></span>)</span> <a href="analmetr.html#R2" title="ANALMETR:pred.2">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E1:46"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="analmetr.html#E6:46"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="analmetr.html#E8:46"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="analmetr.html#E9:46"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="analmetr.html#T16" target="_self" title="ANALMETR:th.16">Th16</a></i>;<br/>
<a NAME="E12:46"/><b>then </b>
<font color="Maroon" title="c11">u2</font>,<font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">u</font>,<font color="Maroon" title="c10">u1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="analmetr.html#E10:46"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="analmetr.html#D3" target="_self" title="ANALMETR:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E13:46"/>
<font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c5">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c6">p1</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T31" target="_self" title="ANALMETR:th.31">Th31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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