<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><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">v</font>, <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">w</font>, <font color="Olive" title="b6">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="b7">p</font>, <font color="Olive" title="b8">p1</font>, <font color="Olive" title="b9">q</font>, <font color="Olive" title="b10">q1</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="b5">w</font>,<font color="Olive" title="b6">y</font></span>)</span>  st <font color="Olive" title="b7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u1</font> &amp; <font color="Olive" title="b9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> &amp; <font color="Olive" title="b10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v1</font> holds <br/>( <font color="Olive" title="b7">p</font>,<font color="Olive" title="b9">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b8">p1</font>,<font color="Olive" title="b10">q1</font> iff <font color="Olive" title="b1">u</font>,<font color="Olive" title="b3">v</font>,<font color="Olive" title="b2">u1</font>,<font color="Olive" title="b4">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Olive" title="b5">w</font>,<font color="Olive" title="b6">y</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">v</font>, <font color="Maroon" title="c5">v1</font>, <font color="Maroon" title="c6">w</font>, <font color="Maroon" title="c7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">q1</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="c6">w</font>,<font color="Maroon" title="c7">y</font></span>)</span>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Olive" title="b2">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Olive" title="b4">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> holds <br/>( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b4">q1</font> iff <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">p1</font>, <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">q1</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="c6">w</font>,<font color="Maroon" title="c7">y</font></span>)</span>; <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="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Maroon" title="c9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> implies ( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</font> iff <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font> ) )</span><br/>





















<span class="kw">assume </span><a NAME="E1:46"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">u</font> &amp; <font color="Maroon" title="c9">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u1</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> &amp; <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">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="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</font> iff <font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font> )</span><br/>

<a NAME="E2:46"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analmetr.html#U1" title="ANALMETR:sel.1">orthogonality</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="c6">w</font>,<font color="Maroon" title="c7">y</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="analmetr.html#D6" target="_self" title="ANALMETR:def.6">Def6</a></span>;<br/>
<div><span class="kw">hereby </span> <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="c4">v</font>,<font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:46_1"/>
<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</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="c4">v</font>,<font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">u9</font>, <font color="Maroon" title="c13">v9</font>, <font color="Maroon" title="c14">u19</font>, <font color="Maroon" title="c15">v19</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:46_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">v</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">u9</font>,<font color="Maroon" title="c13">v9</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:46_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c14">u19</font>,<font color="Maroon" title="c15">v19</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:46_1"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
<font color="Maroon" title="c12">u9</font>,<font color="Maroon" title="c13">v9</font>,<font color="Maroon" title="c14">u19</font>,<font color="Maroon" title="c15">v19</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="analmetr.html#D4" target="_self" title="ANALMETR:def.4">Def4</a></span>;<br/><a NAME="E6:46_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
<font color="Maroon" title="c3">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E7:46_1"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u9</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><span class="kw">hence </span><a NAME="E8:46_1"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:46_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E5:46_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E6:46_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</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">assume </span><a NAME="E4:46"/>
<font color="Maroon" title="c2">u</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c3">u1</font>,<font color="Maroon" title="c5">v1</font> <a href="analmetr.html#R3" title="ANALMETR:pred.3">are_Ort_wrt</a> <font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</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="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</font></span><br/>

<span class="kw">hence </span><a NAME="E5:46"/>
<font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c10">q</font> <a href="analmetr.html#R4" title="ANALMETR:pred.4">_|_</a> <font color="Maroon" title="c9">p1</font>,<font color="Maroon" title="c11">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="analmetr.html#D4" target="_self" title="ANALMETR: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>
