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<b>let </b><font color="Maroon" title="c1">B0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">B0</font> is <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a> iff for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  )</span><br/>

<b>thus </b><a NAME="E1:23"/>
( <font color="Maroon" title="c1">B0</font> is <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a> implies for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</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="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  ) implies <font color="Maroon" title="c1">B0</font> is <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">B0</font> is <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</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="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> </span><br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a>  )</span><br/>

<b>assume </b><a NAME="E2:23_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> </span><br/>

<b>hence </b><a NAME="E3:23_1"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="euclid_7.html#D3" target="_self" title="EUCLID_7:def.3">Def4</a>, <a class="txt" href="#E1:23_1"><i><font color="Green" title="E16">A1</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="euclid_2.html#D3" title="EUCLID_2:def.3">EUCLID_2:def 3</a> <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>

<b>assume </b><a NAME="E2:23"/><i><font color="Green" title="E16">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">B0</font> is <a href="euclid_7.html#V1" title="EUCLID_7:attr.1">R-orthogonal</a></span><br/>

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="euclid_7.html#D3" target="_self" title="EUCLID_7:def.3">EUCLID_7:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> implies <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<b>assume </b><a NAME="E3:23"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B0</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <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"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E4:23"/><b>then </b>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="euclid_2.html#R1" title="EUCLID_2:pred.1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:23"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="euclid_2.html#D3" title="EUCLID_2:def.3">EUCLID_2:def 3</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>
