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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>;<br/><b>let </b><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>;<br/><b>let </b><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>;<br/>





















<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E29">A1</font></i>: 
( <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> )
 ;<br/>

<a NAME="E2:36"/><i><font color="Green" title="E30">A2</font></i>: 
( <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> )
 
<b>by </b><i><a class="ref" href="analmetr.html#D6" target="_self" title="ANALMETR:def.6">Def6</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:36_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>
 ;<br/><a NAME="E2:36_1"/><b>then </b>
<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="c2">u</font>,<font color="Maroon" title="c4">v</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="c3">u1</font>,<font color="Maroon" title="c5">v1</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>  <a href="analmetr.html#K1" title="ANALMETR:func.1">Orthogonality</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="analmetr.html#E1:36"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="analmetr.html#E2:36"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">u'</font>, <font color="Maroon" title="c13">v'</font>, <font color="Maroon" title="c14">u1'</font>, <font color="Maroon" title="c15">v1'</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:36_1"/><i><font color="Green" title="E31">A3</font></i>: 
( <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">u'</font>,<font color="Maroon" title="c13">v'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <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">u1'</font>,<font color="Maroon" title="c15">v1'</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <b>and </b><br/><a NAME="E5:36_1"/><i><font color="Green" title="E32">A4</font></i>: 
<font color="Maroon" title="c12">u'</font>,<font color="Maroon" title="c13">v'</font>,<font color="Maroon" title="c14">u1'</font>,<font color="Maroon" title="c15">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>
 <b>by </b><i><a class="ref" href="analmetr.html#D4" target="_self" title="ANALMETR:def.4">Def4</a></i>;<br/><a NAME="E6:36_1"/>
( <font color="Maroon" title="c2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">u'</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">v'</font> &amp; <font color="Maroon" title="c3">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">u1'</font> &amp; <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">v1'</font> )
 <b>by </b><i><a class="txt" href="analmetr.html#E4:36_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E7:36_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>
 <b>by </b><i><a class="txt" href="analmetr.html#E5:36_1"><i><font color="Green" title="E32">A4</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:36"/>
<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>
 ;<br/>

<a NAME="E5:36"/><b>then </b>
<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="c2">u</font>,<font color="Maroon" title="c4">v</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="c3">u1</font>,<font color="Maroon" title="c5">v1</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>  <a href="analmetr.html#K1" title="ANALMETR:func.1">Orthogonality</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c6">w</font>,<font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="ref" href="analmetr.html#D4" target="_self" title="ANALMETR:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E6:36"/>
<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>
 <b>by </b><i><a class="txt" href="analmetr.html#E1:36"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="analmetr.html#E2:36"><i><font color="Green" title="E30">A2</font></i></a></i>;<br/>


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