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<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of 3,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</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">  for <font color="Olive" title="b1">P</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span></span>)</span><br/>  for <font color="Olive" title="b2">a</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b2">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">N</font> holds <br/><span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="anproj_1.html#K5" title="ANPROJ_1:func.5">ProjectiveSpace</a> <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</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">  for <font color="Olive" title="b1">a</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>  st <font color="Olive" title="b1">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">N</font> holds <br/><span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</font> be   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</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="c3">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">N</font> implies <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E22">A1</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="matrix13.html#K2" title="MATRIX13:func.2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">N</font>
 ; <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"> <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">aN</font> = <font color="Maroon" title="c1">N</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">u</font>, <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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3</span>)</span>, <font color="Maroon" title="c7">uf</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>, <font color="Maroon" title="c8">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a><span class="kw"> such that </span><br/><a NAME="E3:25"/><span class="lab"><font color="Green" title="E23">A2</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c5">u</font>
 <span class="kw">and </span><br/><a NAME="E4:25"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
 not <font color="Maroon" title="c5">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E5:25"/><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">uf</font>
 <span class="kw">and </span><br/><a NAME="E6:25"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c7">uf</font>
 <span class="kw">and </span><br/><a NAME="E7:25"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_8.html#K2" title="ANPROJ_8:func.2">M2F</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">and </span><br/><a NAME="E8:25"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 not <font color="Maroon" title="c6">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E9:25"/><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <font color="Maroon" title="c6">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#D4" onmouseover="rs('anproj_8/D4')" onmouseout="rh()">ANPROJ_8:def 4</a></span>;<br/>
<a NAME="E10:25"/>
<font color="Maroon" title="c5">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 3
 
;<br/>
<a NAME="E11:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c7">uf</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/>
<a NAME="E12:25"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c1">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of 3,1,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="ref" href="finseq_3.html#T153" onmouseover="rs('finseq_3/T153')" onmouseout="rh()">FINSEQ_3:153</a>, <a class="ref" href="anproj_8.html#T91" onmouseover="rs('anproj_8/T91')" onmouseout="rh()">ANPROJ_8:91</a></span>;<br/>
<a NAME="E13:25"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="laplace.html#D9" onmouseover="rs('laplace/D9')" onmouseout="rh()">LAPLACE:def 9</a></span>;<br/>
<a NAME="E14:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:25"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A12</font></span>: <font color="Maroon" title="c8">p</font> = 
<font color="Maroon" title="c1">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:25"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="laplace.html#D9" onmouseover="rs('laplace/D9')" onmouseout="rh()">LAPLACE:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E22">A1</font></span></a>, <a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a>, <a class="txt" href="anproj_9.html#E22"><span class="lab"><font color="Green" title="E20">Lem06</font></span></a></span>
;<br/>
<a NAME="E16:25"/><span class="lab"><font color="Green" title="E34">A13</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p5">(<span class="default"><span class="p0"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p5">(<span class="default"><span class="p0"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p5">(<span class="default"><span class="p0"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E15:25"><span class="lab"><font color="Green" title="E33">A12</font></span></a>, <a class="txt" href="#E14:25"><span class="lab"><font color="Green" title="E32">A11</font></span></a>, <a class="ref" href="anproj_8.html#D2" onmouseover="rs('anproj_8/D2')" onmouseout="rh()">ANPROJ_8:def 2</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> )</span></a><div class="add"><a NAME="E1:25_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K3" title="MATRIX_1:func.3">1.</a> (<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>,3)</span>)</span> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> is 3,1 <a href="matrix_0.html#V2" title="MATRIX_0:attr.2">-size</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a>, <a class="ref" href="anproj_8.html#T91" onmouseover="rs('anproj_8/T91')" onmouseout="rh()">ANPROJ_8:91</a></span>;<br/><a NAME="E2:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A14</font></span>: 
<span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a>  is 3,1 <a href="matrix_0.html#V2" title="MATRIX_0:attr.2">-size</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a>, <a class="ref" href="anproj_8.html#T99" onmouseover="rs('anproj_8/T99')" onmouseout="rh()">ANPROJ_8:99</a></span>;<br/><a NAME="E3:25_2"/><span class="lab"><font color="Green" title="E36">A15</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a>, <a class="ref" href="anproj_8.html#T2" onmouseover="rs('anproj_8/T2')" onmouseout="rh()">ANPROJ_8:2</a></span>;<br/><a NAME="E4:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A16</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,1)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (2,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (2,1)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (3,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (3,1)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>;<br/><a NAME="E5:25_2"/><span class="lab"><font color="Green" title="E38">A17</font></span>: 
<span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:25"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a>, <a class="ref" href="anproj_8.html#T77" onmouseover="rs('anproj_8/T77')" onmouseout="rh()">ANPROJ_8:77</a></span>;<br/><a NAME="E6:25_2"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25_2"><span class="lab"><font color="Green" title="E35">A14</font></span></a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/><a NAME="E7:25_2"/><span class="lab"><font color="Green" title="E40">A19</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></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> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> &amp;  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_3.html#D5" onmouseover="rs('matrix_3/D5')" onmouseout="rh()">MATRIX_3:def 5</a></span>;<br/><a NAME="E8:25_2"/>
<font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> is   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of 3,1,<a href="vectsp_1.html#K2" title="VECTSP_1:func.2">F_Real</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T20" onmouseover="rs('matrix_0/T20')" onmouseout="rh()">MATRIX_0:20</a>, <a class="txt" href="#E7:25_2"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E6:25_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>;<br/><a NAME="E9:25_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A20</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T2" onmouseover="rs('anproj_8/T2')" onmouseout="rh()">ANPROJ_8:2</a>, <a class="ref" href="matrix_0.html#T23" onmouseover="rs('matrix_0/T23')" onmouseout="rh()">MATRIX_0:23</a></span>;<br/><span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (1,1) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="matrprob.html#T14" onmouseover="rs('matrprob/T14')" onmouseout="rh()">MATRPROB:14</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1

;<br/><span class="kw">hence </span><a NAME="E11:25_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E9:25_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="matrprob.html#T14" onmouseover="rs('matrprob/T14')" onmouseout="rh()">MATRPROB:14</a>, <a class="txt" href="#E4:25_2"><span class="lab"><font color="Green" title="E37">A16</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"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> )</span><br/><span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (2,1) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="matrprob.html#T14" onmouseover="rs('matrprob/T14')" onmouseout="rh()">MATRPROB:14</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2

;<br/><span class="kw">hence </span><a NAME="E13:25_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E9:25_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="matrprob.html#T14" onmouseover="rs('matrprob/T14')" onmouseout="rh()">MATRPROB:14</a>, <a class="txt" href="#E4:25_2"><span class="lab"><font color="Green" title="E37">A16</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></span><br/><span class="p1">(<span class="default"><span class="p2"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (3,1) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25_2"><span class="lab"><font color="Green" title="E36">A15</font></span></a>, <a class="ref" href="matrprob.html#T14" onmouseover="rs('matrprob/T14')" onmouseout="rh()">MATRPROB:14</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_2"><span class="lab"><font color="Green" title="E38">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3

;<br/><span class="kw">hence </span><a NAME="E15:25_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="matrix_3.html#K6" title="MATRIX_3:func.6">*</a> <span class="p3">(<span class="default"><span class="p4"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c7">uf</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> <a href="matrix_0.html#K4" title="MATRIX_0:func.4">@</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E9:25_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="matrprob.html#T14" onmouseover="rs('matrprob/T14')" onmouseout="rh()">MATRPROB:14</a>, <a class="txt" href="#E4:25_2"><span class="lab"><font color="Green" title="E37">A16</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/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="lab"><font color="Green" title="E35">A21</font></span>: <font color="Maroon" title="c6">v</font> = 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:25"><span class="lab"><font color="Green" title="E34">A13</font></span></a>, <a class="ref" href="euclid_5.html#D1" onmouseover="rs('euclid_5/D1')" onmouseout="rh()">EUCLID_5:def 1</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#D2" onmouseover="rs('euclid_5/D2')" onmouseout="rh()">EUCLID_5:def 2</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a></span>)</span></span>)</span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#D3" onmouseover="rs('euclid_5/D3')" onmouseout="rh()">EUCLID_5:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T3" onmouseover="rs('euclid_5/T3')" onmouseout="rh()">EUCLID_5:3</a></span>
;<br/>
<a NAME="E19:25"/>
 not <font color="Maroon" title="c3">a</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
;<br/>
<a NAME="E20:25"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c5">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:25"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:25"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="txt" href="#E4:25"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="txt" href="#E8:25"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</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/>


</div>
