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<span class="kw">let </span><font color="Maroon" title="c2">Bn</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Bn</font> is  <a href="euclid_7.html#V3" title="EUCLID_7:attr.3">R-orthonormal</a>  implies <font color="Maroon" title="c2">Bn</font> is  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:86_1_1"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c2">Bn</font> is  <a href="euclid_7.html#V3" title="EUCLID_7:attr.3">R-orthonormal</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">Bn</font> is  <a href="rlvect_3.html#V1" title="RLVECT_3:attr.1">linearly-independent</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">l</font> be    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">Bn</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rlvect_3.html#D1" onmouseover="rs('rlvect_3/D1')" onmouseout="rh()">RLVECT_3:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> or  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:86_1_1"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">v0</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:86_1_1"/><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E5:86_1_1"/><span class="lab"><font color="Green" title="E45">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font>
 <span class="kw">and </span><br/><a NAME="E6:86_1_1"/><span class="lab"><font color="Green" title="E46">A5</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" onmouseover="rs('rlvect_2/D8')" onmouseout="rh()">RLVECT_2:def 8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:86_1_1"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E8:86_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A7</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font>
 
;<br/>
<a NAME="E9:86_1_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v2</font> where <font color="Olive" title="b1">v2</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> : <font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D4" onmouseover="rs('rlvect_2/D4')" onmouseout="rh()">RLVECT_2:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E11:86_1_1"/><span class="lab"><font color="Green" title="E49">A8</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font>
 <span class="kw">and </span><br/><a NAME="E12:86_1_1"/>
<font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">xv</font> = <font color="Maroon" title="c6">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T93" onmouseover="rs('finseq_2/T93')" onmouseout="rh()">FINSEQ_2:93</a></span>;<br/>
<span class="lab"><font color="Green" title="E50">A9</font></span>: <span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">v</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span>) = 
<span class="p1"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><font color="Maroon" title="c7">xv</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:86_1_1"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E6:86_1_1"><span class="lab"><font color="Green" title="E46">A5</font></span></a>, <a class="ref" href="real_ns1.html#D5" onmouseover="rs('real_ns1/D5')" onmouseout="rh()">REAL_NS1:def 5</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T18" onmouseover="rs('euclid_4/T18')" onmouseout="rh()">EUCLID_4:18</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">x0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E16:86_1_1"/><span class="lab"><font color="Green" title="E51">A10</font></span>: 
<font color="Maroon" title="c8">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E17:86_1_1"/><span class="lab"><font color="Green" title="E52">A11</font></span>: 
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:86_1_1"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E5:86_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">nx0</font> = <font color="Maroon" title="c8">x0</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86_1_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a></span>;<br/>
<a NAME="E19:86_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86_1_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E20:86_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86_1_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E21:86_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">v3</font> where <font color="Olive" title="b1">v3</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> : <font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:86_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="ref" href="rlvect_2.html#D4" onmouseover="rs('rlvect_2/D4')" onmouseout="rh()">RLVECT_2:def 4</a></span>;<br/>
<a NAME="E22:86_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A12</font></span>: 
 ex <font color="Olive" title="b1">v3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="funcsdom.html#K10" title="FUNCSDOM:func.10">RealVectSpace</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> st <br/>( <font color="Olive" title="b1">v3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font> &amp; <font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
;<br/>
<a NAME="E23:86_1_1"/><span class="lab"><font color="Green" title="E54">A13</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E24:86_1_1"/><span class="lab"><font color="Green" title="E55">A14</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c3">l</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Bn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D6" onmouseover="rs('rlvect_2/D6')" onmouseout="rh()">RLVECT_2:def 6</a></span>;<br/>
<a NAME="E25:86_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D7" onmouseover="rs('rlvect_2/D7')" onmouseout="rh()">RLVECT_2:def 7</a></span>;<br/>
<a NAME="E26:86_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">nx0</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:86_1_1"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="txt" href="#E23:86_1_1"><span class="lab"><font color="Green" title="E54">A13</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">v</font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="rlvect_2.html#K5" title="RLVECT_2:func.5">(#)</a> <font color="Maroon" title="c5">F</font></span>)</span></span>)</span>) = 
<span class="p1">(<span class="default"><a href="real_ns1.html#K6" title="REAL_NS1:func.6">Euclid_scalar</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">v</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E24:86_1_1"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E8:86_1_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E11:86_1_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="txt" href="#E4:86_1_1"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="txt" href="#E5:86_1_1"><span class="lab"><font color="Green" title="E45">A4</font></span></a>, <a class="txt" href="#E17:86_1_1"><span class="lab"><font color="Green" title="E52">A11</font></span></a>, <a class="ref" href="euclid_7.html#T39" target="_self" onmouseover="rs('euclid_7/T39')" onmouseout="rh()">Th38</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><font color="Maroon" title="c7">xv</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c7">xv</font></span>)</span></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D5" onmouseover="rs('real_ns1/D5')" onmouseout="rh()">REAL_NS1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1"><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">|(</a><span class="default"><font color="Maroon" title="c7">xv</font>,<font color="Maroon" title="c7">xv</font></span><a href="rvsum_1.html#K22" title="RVSUM_1:func.22">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_4.html#T22" onmouseover="rs('euclid_4/T22')" onmouseout="rh()">EUCLID_4:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c7">xv</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_2.html#T4" onmouseover="rs('euclid_2/T4')" onmouseout="rh()">EUCLID_2:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default">1 <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:86_1_1"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E24:86_1_1"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="txt" href="#E8:86_1_1"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="txt" href="#E11:86_1_1"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="euclid_7.html#D4" target="_self" onmouseover="rs('euclid_7/D4')" onmouseout="rh()">Def4</a></span>
<br/>.= 
<font color="Maroon" title="c3">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">nx0</font></span>)</span>

;<br/>
<span class="kw">hence </span><a NAME="E28:86_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:86_1_1"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="txt" href="#E14:86_1_1"><span class="lab"><font color="Green" title="E50">A9</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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