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<span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">ia</font>, <font color="Maroon" title="c8">ib</font>, <font color="Maroon" title="c9">ic</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">  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">N</font> being   <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>  st <font color="Olive" title="b1">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> &amp; <font color="Maroon" title="c7">ia</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c8">ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c9">ic</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Olive" title="b2">N</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"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">ia</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">ib</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c9">ic</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> holds <br/><span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Olive" title="b2">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>  <a href="anproj_1.html#K2" title="ANPROJ_1:func.2">Dir</a> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,1,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c10">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">N</font> being   <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>  st <font color="Maroon" title="c10">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> &amp; <font color="Maroon" title="c7">ia</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c8">ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c9">ic</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Olive" title="b1">N</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"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">ia</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">ib</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c9">ic</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> holds <br/><span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Olive" title="b1">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,1,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c11">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"> ( <font color="Maroon" title="c10">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> &amp; <font color="Maroon" title="c7">ia</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c8">ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c9">ic</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c11">N</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"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">ia</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">ib</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c9">ic</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span> implies <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,1,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E35">A1</font></span>: 
<font color="Maroon" title="c10">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E36">A2</font></span>: 
<font color="Maroon" title="c7">ia</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">a</font>
 <span class="kw">and </span><br/><a NAME="E3:47"/><span class="lab"><font color="Green" title="E37">A3</font></span>: 
<font color="Maroon" title="c8">ib</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">and </span><br/><a NAME="E4:47"/><span class="lab"><font color="Green" title="E38">A4</font></span>: 
<font color="Maroon" title="c9">ic</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E5:47"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<font color="Maroon" title="c11">N</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"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c7">ia</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c8">ib</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c9">ic</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</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"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,1,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">u</font>, <font color="Maroon" title="c13">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="c14">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="c15">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="E7:47"/><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c10">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="c12">u</font>
 <span class="kw">and </span><br/><a NAME="E8:47"/><span class="lab"><font color="Green" title="E41">A16</font></span>: 
 not <font color="Maroon" title="c12">u</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E9:47"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">uf</font>
 <span class="kw">and </span><br/><a NAME="E10:47"/><span class="lab"><font color="Green" title="E43">A18</font></span>: 
<font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c14">uf</font>
 <span class="kw">and </span><br/><a NAME="E11:47"/><span class="lab"><font color="Green" title="E44">A19</font></span>: 
<font color="Maroon" title="c13">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="c15">p</font>
 <span class="kw">and </span><br/><a NAME="E12:47"/><span class="lab"><font color="Green" title="E45">A20</font></span>: 
 not <font color="Maroon" title="c13">v</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E13:47"/><span class="lab"><font color="Green" title="E46">A21</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="c11">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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="c13">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="E14:47"/>
 not <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:47_1"/>
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</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="E2:47_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">a</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Maroon" title="c5">b</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp; <font color="Maroon" title="c6">c</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T4" onmouseover="rs('euclid_5/T4')" onmouseout="rh()">EUCLID_5:4</a>, <a class="ref" href="finseq_1.html#T78" onmouseover="rs('finseq_1/T78')" onmouseout="rh()">FINSEQ_1:78</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:47_1"/>
contradiction
 ; <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>
<a NAME="E15:47"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c12">u</font>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E35">A1</font></span></a>, <a class="txt" href="#E7:47"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="txt" href="#E8:47"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">d</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E17:47"/><span class="lab"><font color="Green" title="E47">A22</font></span>: 
<font color="Maroon" title="c16">d</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 class="kw">and </span><br/><a NAME="E18:47"/><span class="lab"><font color="Green" title="E48">A23</font></span>: 
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</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"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font></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="anproj_1.html#T1" onmouseover="rs('anproj_1/T1')" onmouseout="rh()">ANPROJ_1:1</a></span>;<br/>
<a NAME="E19:47"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c12">u</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"><font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font></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="#E18:47"><span class="lab"><font color="Green" title="E48">A23</font></span></a>, <a class="ref" href="euclid_5.html#T8" onmouseover="rs('euclid_5/T8')" onmouseout="rh()">EUCLID_5:8</a></span>;<br/>
<a NAME="E20:47"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
( <font color="Maroon" title="c14">uf</font> <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="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c14">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c14">uf</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:47"><span class="lab"><font color="Green" title="E49">A24</font></span></a>, <a class="txt" href="#E9:47"><span class="lab"><font color="Green" title="E42">A17</font></span></a></span>;<br/>
<a NAME="E21:47"/>
<font color="Maroon" title="c12">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="E22:47"/><span class="kw">then </span>
<font color="Maroon" title="c12">u</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="ref" href="euclid.html#T22" onmouseover="rs('euclid/T22')" onmouseout="rh()">EUCLID:22</a></span>;<br/>
<a NAME="E23:47"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A26</font></span>: 
<span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c14">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="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font></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="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font></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="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font></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="#E20:47"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="txt" href="#E9:47"><span class="lab"><font color="Green" title="E42">A17</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">Mu</font> = <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c14">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>  as   <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="#E23:47"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="ref" href="anproj_8.html#T4" onmouseover="rs('anproj_8/T4')" onmouseout="rh()">ANPROJ_8:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">z</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> , <font color="Maroon" title="c19">da</font> = <font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c20">db</font> = <font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c21">dc</font> = <font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E52">A27</font></span>: <font color="Maroon" title="c15">p</font> = 
<font color="Maroon" title="c11">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <font color="Maroon" title="c17">Mu</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:47"><span class="lab"><font color="Green" title="E43">A18</font></span></a>, <a class="ref" href="laplace.html#D9" onmouseover="rs('laplace/D9')" onmouseout="rh()">LAPLACE:def 9</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"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></span>)</span></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>,<span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c18">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c18">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></span>)</span></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>,<span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c18">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c18">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></span>)</span></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&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="#E23:47"><span class="lab"><font color="Green" title="E51">A26</font></span></a>, <a class="txt" href="#E5:47"><span class="lab"><font color="Green" title="E39">A5</font></span></a>, <a class="ref" href="anproj_9.html#T7" target="_self" onmouseover="rs('anproj_9/T7')" onmouseout="rh()">Th08</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"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>,<span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span>,<span class="p2"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span>

;<br/>
<a NAME="E27:47"/><span class="lab"><font color="Green" title="E53">A28</font></span>: 
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></span>)</span></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:47_3"/><span class="lab"><font color="Green" title="E54">A29</font></span>: 
( <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></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="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font> &amp; <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></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="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font> &amp; <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></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="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font> )
 
;<br/>
<a NAME="E2:47_3"/><span class="lab"><font color="Green" title="E55">A30</font></span>: 
( <font color="Maroon" title="c15">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c15">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c15">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_0.html#K13" title="MATRIX_0:func.13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:47"><span class="lab"><font color="Green" title="E52">A27</font></span></a></span>;<br/>
<a NAME="E3:47_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">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="#E26:47"><span class="lab"><font color="Green" title="E52">A27</font></span></a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:47_3"/>
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font></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="txt" href="#E1:47_3"><span class="lab"><font color="Green" title="E54">A29</font></span></a>, <a class="txt" href="#E2:47_3"><span class="lab"><font color="Green" title="E55">A30</font></span></a>, <a class="txt" href="#E11:47"><span class="lab"><font color="Green" title="E44">A19</font></span></a>, <a class="ref" href="anproj_8.html#D2" onmouseover="rs('anproj_8/D2')" onmouseout="rh()">ANPROJ_8:def 2</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>
<a NAME="E28:47"/><span class="lab"><font color="Green" title="E54">A31</font></span>: 
(  not <font color="Maroon" title="c4">a</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c5">b</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c6">c</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  )
 
;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="47_4"><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"> ( <font color="Maroon" title="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font> &amp; <font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font> &amp; <font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font> )</span></a><div class="add"><span class="kw">thus </span><font color="Maroon" title="c7">ia</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c19">da</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E36">A2</font></span></a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:47"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="ref" href="xcmplx_1.html#T106" onmouseover="rs('xcmplx_1/T106')" onmouseout="rh()">XCMPLX_1:106</a></span>
<br/>.= 
<font color="Maroon" title="c16">d</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"> ( <font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font> &amp; <font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font> )</span><br/><span class="kw">thus </span><font color="Maroon" title="c8">ib</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c20">db</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47"><span class="lab"><font color="Green" title="E37">A3</font></span></a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:47"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="ref" href="xcmplx_1.html#T106" onmouseover="rs('xcmplx_1/T106')" onmouseout="rh()">XCMPLX_1:106</a></span>
<br/>.= 
<font color="Maroon" title="c16">d</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"> <font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">d</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c9">ic</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c21">dc</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E38">A4</font></span></a></span>
<br/>.= 
1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c16">d</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:47"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="ref" href="xcmplx_1.html#T106" onmouseover="rs('xcmplx_1/T106')" onmouseout="rh()">XCMPLX_1:106</a></span>
<br/>.= 
<font color="Maroon" title="c16">d</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"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><font color="Maroon" title="c13">v</font> = 
<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="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">d</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1</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="txt" href="#E27:47"><span class="lab"><font color="Green" title="E53">A28</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c16">d</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">1,1,1</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/>
<a NAME="E31:47"/><span class="kw">then </span>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c13">v</font>,<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,1,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:47"><span class="lab"><font color="Green" title="E47">A22</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="E32:47"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c11">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">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> <span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,1,1</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="anproj_9.html#T10" target="_self" onmouseover="rs('anproj_9/T10')" onmouseout="rh()">Th11</a>, <a class="txt" href="#E12:47"><span class="lab"><font color="Green" title="E45">A20</font></span></a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a>, <a class="txt" href="#E13:47"><span class="lab"><font color="Green" title="E46">A21</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>
