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<span class="kw">let </span><font color="Maroon" title="c4">P1</font>, <font color="Maroon" title="c5">P2</font>, <font color="Maroon" title="c6">P3</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"> (  not <font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c5">P2</font>,<font color="Maroon" title="c6">P3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</a>  implies  ex <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 <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="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  &amp; <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="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  &amp; <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="c6">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 not <font color="Maroon" title="c4">P1</font>,<font color="Maroon" title="c5">P2</font>,<font color="Maroon" title="c6">P3</font> <a href="collsp.html#R1" title="COLLSP:pred.1">are_collinear</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">  ex <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 <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="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  &amp; <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="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  &amp; <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="c6">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">u1</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><span class="kw"> such that </span><br/><a NAME="E3:42"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 not <font color="Maroon" title="c7">u1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E4:42"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c4">P1</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="c7">u1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T26" onmouseover="rs('anproj_1/T26')" onmouseout="rh()">ANPROJ_1:26</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">u2</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><span class="kw"> such that </span><br/><a NAME="E6:42"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
 not <font color="Maroon" title="c8">u2</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E7:42"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c5">P2</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="c8">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T26" onmouseover="rs('anproj_1/T26')" onmouseout="rh()">ANPROJ_1:26</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">u3</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><span class="kw"> such that </span><br/><a NAME="E9:42"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
 not <font color="Maroon" title="c9">u3</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E10:42"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c6">P3</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="c9">u3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_1.html#T26" onmouseover="rs('anproj_1/T26')" onmouseout="rh()">ANPROJ_1:26</a></span>;<br/>
<a NAME="E11:42"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<span class="p1"><a href="euclid_5.html#K6" title="EUCLID_5:func.6">|{</a><span class="default"><font color="Maroon" title="c7">u1</font>,<font color="Maroon" title="c8">u2</font>,<font color="Maroon" title="c9">u3</font></span><a href="euclid_5.html#K6" title="EUCLID_5:func.6">}|</a></span> <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">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T43" onmouseover="rs('anproj_8/T43')" onmouseout="rh()">ANPROJ_8:43</a>, <a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="anproj_2.html#T23" onmouseover="rs('anproj_2/T23')" onmouseout="rh()">ANPROJ_2:23</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">pf</font> = <font color="Maroon" title="c7">u1</font>, <font color="Maroon" title="c11">qf</font> = <font color="Maroon" title="c8">u2</font>, <font color="Maroon" title="c12">rf</font> = <font color="Maroon" title="c9">u3</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T145" onmouseover="rs('rvsum_1/T145')" onmouseout="rh()">RVSUM_1:145</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">N</font> being   <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><span class="kw"> such that </span><br/><a NAME="E14:42"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c13">N</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">and </span><br/><a NAME="E15:42"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c13">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a>
 <span class="kw">and </span><br/><a NAME="E16:42"/><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c13">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a>
 <span class="kw">and </span><br/><a NAME="E17:42"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c13">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="anproj_8.html#T94" onmouseover="rs('anproj_8/T94')" onmouseout="rh()">ANPROJ_8:94</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">N</font> = <font color="Maroon" title="c13">N</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E38">A9</font></span></a></span>;<br/>
<a NAME="E19:42"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">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="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a>  &amp; <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  &amp; <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E20:42"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</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"> ( <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a>  &amp; <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c15">nu1</font>, <font color="Maroon" title="c16">nv1</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="c17">u1f</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="c18">p1</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="E2:42_1"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c4">P1</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="c15">nu1</font>
 <span class="kw">and </span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
 not <font color="Maroon" title="c15">nu1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E4:42_1"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
<font color="Maroon" title="c15">nu1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u1f</font>
 <span class="kw">and </span><br/><a NAME="E5:42_1"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
<font color="Maroon" title="c18">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c17">u1f</font>
 <span class="kw">and </span><br/><a NAME="E6:42_1"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<font color="Maroon" title="c16">nv1</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="c18">p1</font>
 <span class="kw">and </span><br/><a NAME="E7:42_1"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
 not <font color="Maroon" title="c16">nv1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E8:42_1"/><span class="lab"><font color="Green" title="E49">A20</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="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">P1</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="c16">nv1</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="E9:42_1"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c7">u1</font>,<font color="Maroon" title="c15">nu1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E44">A15</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="c19">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:42_1"/><span class="lab"><font color="Green" title="E50">A21</font></span>: 
<font color="Maroon" title="c19">a</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="E12:42_1"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
<font color="Maroon" title="c7">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">nu1</font>
 <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">b</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c19">a</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E14:42_1"/>
<font color="Maroon" title="c15">nu1</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="E15:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A23</font></span>: 
<font color="Maroon" title="c15">nu1</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 href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u1f</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">nu1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="ref" href="anproj_8.html#T75" onmouseover="rs('anproj_8/T75')" onmouseout="rh()">ANPROJ_8:75</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E53">A24</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="c17">u1f</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#K1" title="MATRIX_0:func.1">width</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u1f</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T29" onmouseover="rs('matrix_0/T29')" onmouseout="rh()">MATRIX_0:29</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">nu1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T75" onmouseover="rs('anproj_8/T75')" onmouseout="rh()">ANPROJ_8:75</a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>
;<br/>
<a NAME="E18:42_1"/><span class="lab"><font color="Green" title="E54">A25</font></span>: 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c14">N</font> <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="c17">u1f</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_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a>, <a class="txt" href="#E17:42_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E55">A26</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="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c17">u1f</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</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="c17">u1f</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="laplace.html#D9" onmouseover="rs('laplace/D9')" onmouseout="rh()">LAPLACE:def 9</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:42_1"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E20:42_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#D1" onmouseover="rs('euclid_8/D1')" onmouseout="rh()">EUCLID_8:def 1</a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</a></span>;<br/>
<a NAME="E21:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A27</font></span>: 
 <a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</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"><a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <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"><a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <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"><a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <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="ref" href="anproj_8.html#D1" onmouseover="rs('anproj_8/D1')" onmouseout="rh()">ANPROJ_8:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">s1</font> = <font color="Maroon" title="c17">u1f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c22">s2</font> = <font color="Maroon" title="c17">u1f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2, <font color="Maroon" title="c23">s3</font> = <font color="Maroon" title="c17">u1f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 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="kw">reconsider </span><font color="Maroon" title="c24">t1</font> = <font color="Maroon" title="c7">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c25">t2</font> = <font color="Maroon" title="c7">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2, <font color="Maroon" title="c26">t3</font> = <font color="Maroon" title="c7">u1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 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/>
<a NAME="E24:42_1"/><span class="lab"><font color="Green" title="E57">A28</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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> <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"><a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E21:42_1"><span class="lab"><font color="Green" title="E56">A27</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A29</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<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="euclid_5.html#K4" title="EUCLID_5:func.4">]|</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="#E24:42_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="euclid_8.html#D1" onmouseover="rs('euclid_8/D1')" onmouseout="rh()">EUCLID_8:def 1</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<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="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#D1" onmouseover="rs('euclid_5/D1')" onmouseout="rh()">EUCLID_5:def 1</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E59">A30</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<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="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="euclid_8.html#D1" onmouseover="rs('euclid_8/D1')" onmouseout="rh()">EUCLID_8:def 1</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<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="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> 
<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/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E60">A31</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<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="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="euclid_8.html#D1" onmouseover="rs('euclid_8/D1')" onmouseout="rh()">EUCLID_8:def 1</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default">1,<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="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> 
<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/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<a NAME="E28:42_1"/>
( <font color="Maroon" title="c18">p1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u1f</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)</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c18">p1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u1f</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)</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c18">p1</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u1f</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)</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="#E4:42_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E14"><span class="lab"><font color="Green" title="E13">Lem02</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">p111</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p1</font> <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, <font color="Maroon" title="c28">p121</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p1</font> <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, <font color="Maroon" title="c29">p131</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p1</font> <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 as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E30:42_1"/><span class="lab"><font color="Green" title="E61">A32</font></span>: 
( <font color="Maroon" title="c27">p111</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,1)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c10">pf</font></span>)</span> &amp; <font color="Maroon" title="c28">p121</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,2)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c10">pf</font></span>)</span> &amp; <font color="Maroon" title="c29">p131</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,3)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c10">pf</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="#E4:42_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="txt" href="anproj_9.html#E16"><span class="lab"><font color="Green" title="E15">Lem04</font></span></a></span>;<br/>
<a NAME="E31:42_1"/><span class="lab"><font color="Green" title="E62">A33</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c30">pp1</font>, <font color="Maroon" title="c31">pp2</font>, <font color="Maroon" title="c32">pp3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:42_1_7"/><span class="lab"><font color="Green" title="E63">A34</font></span>: 
<font color="Maroon" title="c30">pp1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p1</font> <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">and </span><br/><a NAME="E3:42_1_7"/><span class="lab"><font color="Green" title="E64">A35</font></span>: 
<font color="Maroon" title="c31">pp2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p1</font> <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">and </span><br/><a NAME="E4:42_1_7"/><span class="lab"><font color="Green" title="E65">A36</font></span>: 
<font color="Maroon" title="c32">pp3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p1</font> <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">and </span><br/><a NAME="E5:42_1_7"/><span class="lab"><font color="Green" title="E66">A37</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p1</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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</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="#E5:42_1"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="txt" href="#E19:42_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="ref" href="anproj_8.html#D3" onmouseover="rs('anproj_8/D3')" onmouseout="rh()">ANPROJ_8:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_7_1"><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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:42_1_7_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="anproj_8.html#T78" onmouseover="rs('anproj_8/T78')" onmouseout="rh()">ANPROJ_8:78</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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/><span class="kw">thus </span><a NAME="E2:42_1_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_7_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:42_1_7_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,1)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c10">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/>
<a NAME="E2:42_1_7_1_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c27">p111</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a>, <a class="txt" href="#E25:42_1"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="txt" href="#E30:42_1"><span class="lab"><font color="Green" title="E61">A32</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42_1_7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_7"><span class="lab"><font color="Green" title="E63">A34</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</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">thus </span><a NAME="E3:42_1_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_7_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:42_1_7_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,2)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c10">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:42_1_7_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_7"><span class="lab"><font color="Green" title="E64">A35</font></span></a>, <a class="txt" href="#E24:42_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="txt" href="#E26:42_1"><span class="lab"><font color="Green" title="E59">A30</font></span></a>, <a class="txt" href="#E30:42_1"><span class="lab"><font color="Green" title="E61">A32</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><a NAME="E4:42_1_7_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,3)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c10">pf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_1_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_7"><span class="lab"><font color="Green" title="E65">A36</font></span></a>, <a class="txt" href="#E24:42_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="txt" href="#E30:42_1"><span class="lab"><font color="Green" title="E61">A32</font></span></a>, <a class="txt" href="#E27:42_1"><span class="lab"><font color="Green" title="E60">A31</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">hence </span><a NAME="E7:42_1_7"/>
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c10">pf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_7"><span class="lab"><font color="Green" title="E66">A37</font></span></a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</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="E32:42_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E63">A38</font></span>:  <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a>  = 
 <a href="anproj_8.html#K2" title="ANPROJ_8:func.2">M2F</a> <span class="p1">(<span class="default"><a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K2" title="EUCLID_8:func.2">&lt;e1&gt;</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T86" onmouseover="rs('anproj_8/T86')" onmouseout="rh()">ANPROJ_8:86</a></span>
<br/>.= 
<font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E31:42_1"><span class="lab"><font color="Green" title="E62">A33</font></span></a>, <a class="txt" href="#E15:42"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E19:42_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="anproj_8.html#T83" onmouseover="rs('anproj_8/T83')" onmouseout="rh()">ANPROJ_8:83</a></span>
;<br/>
<a NAME="E34:42_1"/>
(  not <font color="Maroon" title="c16">nv1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c16">nv1</font>,<font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_1"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="ref" href="anproj_9.html#T3" target="_self" onmouseover="rs('anproj_9/T3')" onmouseout="rh()">Th04</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="E35:42_1"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K5" title="ANPROJ_9:func.5">Dir100</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42_1"><span class="lab"><font color="Green" title="E63">A38</font></span></a>, <a class="ref" href="euclid_8.html#D1" onmouseover="rs('euclid_8/D1')" onmouseout="rh()">EUCLID_8:def 1</a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a>, <a class="txt" href="#E8:42_1"><span class="lab"><font color="Green" title="E49">A20</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">thus </span><a NAME="E21:42"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</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"> <span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c15">nu2</font>, <font color="Maroon" title="c16">nv2</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="c17">u2f</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="c18">p2</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="E2:42_2"/><span class="lab"><font color="Green" title="E43">A39</font></span>: 
<font color="Maroon" title="c5">P2</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="c15">nu2</font>
 <span class="kw">and </span><br/><a NAME="E3:42_2"/><span class="lab"><font color="Green" title="E44">A40</font></span>: 
 not <font color="Maroon" title="c15">nu2</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E4:42_2"/><span class="lab"><font color="Green" title="E45">A41</font></span>: 
<font color="Maroon" title="c15">nu2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u2f</font>
 <span class="kw">and </span><br/><a NAME="E5:42_2"/><span class="lab"><font color="Green" title="E46">A42</font></span>: 
<font color="Maroon" title="c18">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c17">u2f</font>
 <span class="kw">and </span><br/><a NAME="E6:42_2"/><span class="lab"><font color="Green" title="E47">A43</font></span>: 
<font color="Maroon" title="c16">nv2</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="c18">p2</font>
 <span class="kw">and </span><br/><a NAME="E7:42_2"/><span class="lab"><font color="Green" title="E48">A44</font></span>: 
 not <font color="Maroon" title="c16">nv2</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E8:42_2"/><span class="lab"><font color="Green" title="E49">A45</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="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">P2</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="c16">nv2</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="E9:42_2"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c8">u2</font>,<font color="Maroon" title="c15">nu2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="txt" href="#E3:42_2"><span class="lab"><font color="Green" title="E44">A40</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="c19">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:42_2"/><span class="lab"><font color="Green" title="E50">A46</font></span>: 
<font color="Maroon" title="c19">a</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="E12:42_2"/><span class="lab"><font color="Green" title="E51">A47</font></span>: 
<font color="Maroon" title="c8">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">nu2</font>
 <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">b</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c19">a</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E14:42_2"/>
<font color="Maroon" title="c15">nu2</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="E15:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A48</font></span>: 
<font color="Maroon" title="c15">nu2</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 href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u2f</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">nu2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="ref" href="anproj_8.html#T75" onmouseover="rs('anproj_8/T75')" onmouseout="rh()">ANPROJ_8:75</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_2"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E53">A49</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="c17">u2f</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#K1" title="MATRIX_0:func.1">width</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u2f</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T29" onmouseover="rs('matrix_0/T29')" onmouseout="rh()">MATRIX_0:29</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">nu2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T75" onmouseover="rs('anproj_8/T75')" onmouseout="rh()">ANPROJ_8:75</a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_2"><span class="lab"><font color="Green" title="E52">A48</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>
;<br/>
<a NAME="E18:42_2"/><span class="lab"><font color="Green" title="E54">A50</font></span>: 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c14">N</font> <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="c17">u2f</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_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a>, <a class="txt" href="#E17:42_2"><span class="lab"><font color="Green" title="E53">A49</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E55">A51</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="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c17">u2f</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</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="c17">u2f</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="laplace.html#D9" onmouseover="rs('laplace/D9')" onmouseout="rh()">LAPLACE:def 9</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:42_2"><span class="lab"><font color="Green" title="E54">A50</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E20:42_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#D2" onmouseover="rs('euclid_8/D2')" onmouseout="rh()">EUCLID_8:def 2</a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</a></span>;<br/>
<a NAME="E21:42_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A52</font></span>: 
 <a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</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"><a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <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"><a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <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"><a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <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="ref" href="anproj_8.html#D1" onmouseover="rs('anproj_8/D1')" onmouseout="rh()">ANPROJ_8:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">s1</font> = <font color="Maroon" title="c17">u2f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c22">s2</font> = <font color="Maroon" title="c17">u2f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2, <font color="Maroon" title="c23">s3</font> = <font color="Maroon" title="c17">u2f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 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="kw">reconsider </span><font color="Maroon" title="c24">t1</font> = <font color="Maroon" title="c8">u2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c25">t2</font> = <font color="Maroon" title="c8">u2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2, <font color="Maroon" title="c26">t3</font> = <font color="Maroon" title="c8">u2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 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/>
<a NAME="E24:42_2"/><span class="lab"><font color="Green" title="E57">A53</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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> <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"><a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E21:42_2"><span class="lab"><font color="Green" title="E56">A52</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A54</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</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="#E24:42_2"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="ref" href="euclid_8.html#D2" onmouseover="rs('euclid_8/D2')" onmouseout="rh()">EUCLID_8:def 2</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#D1" onmouseover="rs('euclid_5/D1')" onmouseout="rh()">EUCLID_5:def 1</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E59">A55</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_2"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="ref" href="euclid_8.html#D2" onmouseover="rs('euclid_8/D2')" onmouseout="rh()">EUCLID_8:def 2</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> 
<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/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E60">A56</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_2"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="ref" href="euclid_8.html#D2" onmouseover="rs('euclid_8/D2')" onmouseout="rh()">EUCLID_8:def 2</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1,<a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> 
<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/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<a NAME="E28:42_2"/>
( <font color="Maroon" title="c18">p2</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u2f</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)</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c18">p2</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u2f</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)</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c18">p2</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u2f</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)</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="#E4:42_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E14"><span class="lab"><font color="Green" title="E13">Lem02</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">p211</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p2</font> <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, <font color="Maroon" title="c28">p221</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p2</font> <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, <font color="Maroon" title="c29">p231</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p2</font> <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 as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E30:42_2"/><span class="lab"><font color="Green" title="E61">A57</font></span>: 
( <font color="Maroon" title="c27">p211</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,1)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c11">qf</font></span>)</span> &amp; <font color="Maroon" title="c28">p221</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,2)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c11">qf</font></span>)</span> &amp; <font color="Maroon" title="c29">p231</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,3)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c11">qf</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E45">A41</font></span></a>, <a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E11:42_2"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="txt" href="#E12:42_2"><span class="lab"><font color="Green" title="E51">A47</font></span></a>, <a class="txt" href="anproj_9.html#E16"><span class="lab"><font color="Green" title="E15">Lem04</font></span></a></span>;<br/>
<a NAME="E31:42_2"/><span class="lab"><font color="Green" title="E62">A58</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c30">pp1</font>, <font color="Maroon" title="c31">pp2</font>, <font color="Maroon" title="c32">pp3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:42_2_7"/><span class="lab"><font color="Green" title="E63">A59</font></span>: 
<font color="Maroon" title="c30">pp1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p2</font> <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">and </span><br/><a NAME="E3:42_2_7"/><span class="lab"><font color="Green" title="E64">A60</font></span>: 
<font color="Maroon" title="c31">pp2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p2</font> <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">and </span><br/><a NAME="E4:42_2_7"/><span class="lab"><font color="Green" title="E65">A61</font></span>: 
<font color="Maroon" title="c32">pp3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p2</font> <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">and </span><br/><a NAME="E5:42_2_7"/><span class="lab"><font color="Green" title="E66">A62</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p2</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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</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="#E5:42_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="txt" href="#E19:42_2"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="ref" href="anproj_8.html#D3" onmouseover="rs('anproj_8/D3')" onmouseout="rh()">ANPROJ_8:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2_7_1"><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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:42_2_7_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="anproj_8.html#T78" onmouseover="rs('anproj_8/T78')" onmouseout="rh()">ANPROJ_8:78</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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/><a NAME="E2:42_2_7_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,1)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c11">qf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2_7"><span class="lab"><font color="Green" title="E63">A59</font></span></a>, <a class="txt" href="#E24:42_2"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="txt" href="#E30:42_2"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="txt" href="#E25:42_2"><span class="lab"><font color="Green" title="E58">A54</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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/><span class="kw">thus </span><a NAME="E4:42_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2_7_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:42_2_7_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,2)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c11">qf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/>
<a NAME="E2:42_2_7_1_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c28">p221</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_2"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a>, <a class="txt" href="#E26:42_2"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="txt" href="#E30:42_2"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="txt" href="#E11:42_2"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42_2_7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_2_7"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</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="E5:42_2_7_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,3)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c11">qf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_2_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2_7"><span class="lab"><font color="Green" title="E65">A61</font></span></a>, <a class="txt" href="#E24:42_2"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="txt" href="#E30:42_2"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="txt" href="#E27:42_2"><span class="lab"><font color="Green" title="E60">A56</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">hence </span><a NAME="E7:42_2_7"/>
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c11">qf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_2_7"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</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="E32:42_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E63">A63</font></span>:  <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a>  = 
 <a href="anproj_8.html#K2" title="ANPROJ_8:func.2">M2F</a> <span class="p1">(<span class="default"><a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K3" title="EUCLID_8:func.3">&lt;e2&gt;</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T86" onmouseover="rs('anproj_8/T86')" onmouseout="rh()">ANPROJ_8:86</a></span>
<br/>.= 
<font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_2"><span class="lab"><font color="Green" title="E47">A43</font></span></a>, <a class="txt" href="#E31:42_2"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="txt" href="#E19:42_2"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E46">A42</font></span></a>, <a class="ref" href="anproj_8.html#T83" onmouseover="rs('anproj_8/T83')" onmouseout="rh()">ANPROJ_8:83</a></span>
;<br/>
<a NAME="E34:42_2"/>
(  not <font color="Maroon" title="c16">nv2</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv2</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c16">nv2</font>,<font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_2"><span class="lab"><font color="Green" title="E48">A44</font></span></a>, <a class="txt" href="#E11:42_2"><span class="lab"><font color="Green" title="E50">A46</font></span></a>, <a class="ref" href="anproj_9.html#T3" target="_self" onmouseover="rs('anproj_9/T3')" onmouseout="rh()">Th04</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="E35:42_2"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K6" title="ANPROJ_9:func.6">Dir010</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42_2"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="ref" href="euclid_8.html#D2" onmouseover="rs('euclid_8/D2')" onmouseout="rh()">EUCLID_8:def 2</a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a>, <a class="txt" href="#E8:42_2"><span class="lab"><font color="Green" title="E49">A45</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">thus </span><a NAME="E22:42"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c15">nu3</font>, <font color="Maroon" title="c16">nv3</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="c17">u3f</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="c18">p3</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="E2:42_3"/><span class="lab"><font color="Green" title="E43">A64</font></span>: 
<font color="Maroon" title="c6">P3</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="c15">nu3</font>
 <span class="kw">and </span><br/><a NAME="E3:42_3"/><span class="lab"><font color="Green" title="E44">A65</font></span>: 
 not <font color="Maroon" title="c15">nu3</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E4:42_3"/><span class="lab"><font color="Green" title="E45">A66</font></span>: 
<font color="Maroon" title="c15">nu3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u3f</font>
 <span class="kw">and </span><br/><a NAME="E5:42_3"/><span class="lab"><font color="Green" title="E46">A67</font></span>: 
<font color="Maroon" title="c18">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c17">u3f</font>
 <span class="kw">and </span><br/><a NAME="E6:42_3"/><span class="lab"><font color="Green" title="E47">A68</font></span>: 
<font color="Maroon" title="c16">nv3</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="c18">p3</font>
 <span class="kw">and </span><br/><a NAME="E7:42_3"/><span class="lab"><font color="Green" title="E48">A69</font></span>: 
 not <font color="Maroon" title="c16">nv3</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">and </span><br/><a NAME="E8:42_3"/><span class="lab"><font color="Green" title="E49">A70</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="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">P3</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="c16">nv3</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="E9:42_3"/>
 <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c9">u3</font>,<font color="Maroon" title="c15">nu3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E2:42_3"><span class="lab"><font color="Green" title="E43">A64</font></span></a>, <a class="txt" href="#E3:42_3"><span class="lab"><font color="Green" title="E44">A65</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="c19">a</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E11:42_3"/><span class="lab"><font color="Green" title="E50">A71</font></span>: 
<font color="Maroon" title="c19">a</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="E12:42_3"/><span class="lab"><font color="Green" title="E51">A72</font></span>: 
<font color="Maroon" title="c9">u3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">nu3</font>
 <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">b</font> = 1 <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <font color="Maroon" title="c19">a</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E14:42_3"/>
<font color="Maroon" title="c15">nu3</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="E15:42_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A73</font></span>: 
<font color="Maroon" title="c15">nu3</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 href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u3f</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">nu3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_3"><span class="lab"><font color="Green" title="E45">A66</font></span></a>, <a class="ref" href="anproj_8.html#T75" onmouseover="rs('anproj_8/T75')" onmouseout="rh()">ANPROJ_8:75</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_3"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E53">A74</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="c17">u3f</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#K1" title="MATRIX_0:func.1">width</a> <span class="p1"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u3f</font></span><a href="laplace.html#K10" title="LAPLACE:func.10">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T29" onmouseover="rs('matrix_0/T29')" onmouseout="rh()">MATRIX_0:29</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">nu3</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T75" onmouseover="rs('anproj_8/T75')" onmouseout="rh()">ANPROJ_8:75</a>, <a class="txt" href="#E4:42_3"><span class="lab"><font color="Green" title="E45">A66</font></span></a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_3"><span class="lab"><font color="Green" title="E52">A73</font></span></a>, <a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>
;<br/>
<a NAME="E18:42_3"/><span class="lab"><font color="Green" title="E54">A75</font></span>: 
 <a href="matrix_0.html#K1" title="MATRIX_0:func.1">width</a> <font color="Maroon" title="c14">N</font> <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="c17">u3f</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_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a>, <a class="txt" href="#E17:42_3"><span class="lab"><font color="Green" title="E53">A74</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E55">A76</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="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c17">u3f</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</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="c17">u3f</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="laplace.html#D9" onmouseover="rs('laplace/D9')" onmouseout="rh()">LAPLACE:def 9</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">N</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:42_3"><span class="lab"><font color="Green" title="E54">A75</font></span></a>, <a class="ref" href="matrix_3.html#D4" onmouseover="rs('matrix_3/D4')" onmouseout="rh()">MATRIX_3:def 4</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_0.html#T24" onmouseover="rs('matrix_0/T24')" onmouseout="rh()">MATRIX_0:24</a></span>
;<br/>
<a NAME="E20:42_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#D3" onmouseover="rs('euclid_8/D3')" onmouseout="rh()">EUCLID_8:def 3</a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</a></span>;<br/>
<a NAME="E21:42_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A77</font></span>: 
 <a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</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"><a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <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"><a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <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"><a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <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="ref" href="anproj_8.html#D1" onmouseover="rs('anproj_8/D1')" onmouseout="rh()">ANPROJ_8:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c21">s1</font> = <font color="Maroon" title="c17">u3f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c22">s2</font> = <font color="Maroon" title="c17">u3f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2, <font color="Maroon" title="c23">s3</font> = <font color="Maroon" title="c17">u3f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 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="kw">reconsider </span><font color="Maroon" title="c24">t1</font> = <font color="Maroon" title="c9">u3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1, <font color="Maroon" title="c25">t2</font> = <font color="Maroon" title="c9">u3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2, <font color="Maroon" title="c26">t3</font> = <font color="Maroon" title="c9">u3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 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/>
<a NAME="E24:42_3"/><span class="lab"><font color="Green" title="E57">A78</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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> <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"><a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E21:42_3"><span class="lab"><font color="Green" title="E56">A77</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A79</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</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>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</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="#E24:42_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="ref" href="euclid_8.html#D3" onmouseover="rs('euclid_8/D3')" onmouseout="rh()">EUCLID_8:def 3</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</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>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#D1" onmouseover="rs('euclid_5/D1')" onmouseout="rh()">EUCLID_5:def 1</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E59">A80</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</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>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="ref" href="euclid_8.html#D3" onmouseover="rs('euclid_8/D3')" onmouseout="rh()">EUCLID_8:def 3</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</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>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> 
<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/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<span class="lab"><font color="Green" title="E60">A81</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</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>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="ref" href="euclid_8.html#D3" onmouseover="rs('euclid_8/D3')" onmouseout="rh()">EUCLID_8:def 3</a></span>
<br/>.= 
<span class="p1"><a href="euclid_5.html#K4" title="EUCLID_5:func.4">|[</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>,1</span><a href="euclid_5.html#K4" title="EUCLID_5:func.4">]|</a></span> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> 
<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/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T2" onmouseover="rs('euclid_5/T2')" onmouseout="rh()">EUCLID_5:2</a></span>
;<br/>
<a NAME="E28:42_3"/>
( <font color="Maroon" title="c18">p3</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u3f</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)</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c18">p3</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u3f</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)</span>)</span></span><a href="matrix_0.html#K13" title="MATRIX_0:func.13">*&gt;</a></span> &amp; <font color="Maroon" title="c18">p3</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"><span class="p3">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="matrix_3.html#K4" title="MATRIX_3:func.4">*</a> <span class="p4">(<span class="default"><span class="p5"><a href="laplace.html#K10" title="LAPLACE:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c17">u3f</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)</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="#E4:42_3"><span class="lab"><font color="Green" title="E45">A66</font></span></a>, <a class="txt" href="#E5:42_3"><span class="lab"><font color="Green" title="E46">A67</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E14"><span class="lab"><font color="Green" title="E13">Lem02</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">p311</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p3</font> <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, <font color="Maroon" title="c28">p321</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p3</font> <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, <font color="Maroon" title="c29">p331</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">p3</font> <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 as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<a NAME="E30:42_3"/><span class="lab"><font color="Green" title="E61">A82</font></span>: 
( <font color="Maroon" title="c27">p311</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,1)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c12">rf</font></span>)</span> &amp; <font color="Maroon" title="c28">p321</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,2)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c12">rf</font></span>)</span> &amp; <font color="Maroon" title="c29">p331</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,3)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c12">rf</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="#E4:42_3"><span class="lab"><font color="Green" title="E45">A66</font></span></a>, <a class="txt" href="#E5:42_3"><span class="lab"><font color="Green" title="E46">A67</font></span></a>, <a class="txt" href="#E11:42_3"><span class="lab"><font color="Green" title="E50">A71</font></span></a>, <a class="txt" href="#E12:42_3"><span class="lab"><font color="Green" title="E51">A72</font></span></a>, <a class="txt" href="anproj_9.html#E16"><span class="lab"><font color="Green" title="E15">Lem04</font></span></a></span>;<br/>
<a NAME="E31:42_3"/><span class="lab"><font color="Green" title="E62">A83</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c30">pp1</font>, <font color="Maroon" title="c31">pp2</font>, <font color="Maroon" title="c32">pp3</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a><span class="kw"> such that </span><br/><a NAME="E2:42_3_7"/><span class="lab"><font color="Green" title="E63">A84</font></span>: 
<font color="Maroon" title="c30">pp1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p3</font> <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">and </span><br/><a NAME="E3:42_3_7"/><span class="lab"><font color="Green" title="E64">A85</font></span>: 
<font color="Maroon" title="c31">pp2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p3</font> <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">and </span><br/><a NAME="E4:42_3_7"/><span class="lab"><font color="Green" title="E65">A86</font></span>: 
<font color="Maroon" title="c32">pp3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">p3</font> <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">and </span><br/><a NAME="E5:42_3_7"/><span class="lab"><font color="Green" title="E66">A87</font></span>: 
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p3</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#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</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="#E5:42_3"><span class="lab"><font color="Green" title="E46">A67</font></span></a>, <a class="txt" href="#E19:42_3"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="ref" href="anproj_8.html#D3" onmouseover="rs('anproj_8/D3')" onmouseout="rh()">ANPROJ_8:def 3</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3_7_1"><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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span></a><div class="add"><span class="kw">thus </span><a NAME="E1:42_3_7_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E19:42"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="anproj_8.html#T78" onmouseover="rs('anproj_8/T78')" onmouseout="rh()">ANPROJ_8:78</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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/><a NAME="E2:42_3_7_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,1)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c12">rf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_3_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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> <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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c30">pp1</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_3_7"><span class="lab"><font color="Green" title="E63">A84</font></span></a>, <a class="txt" href="#E24:42_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="txt" href="#E30:42_3"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="txt" href="#E25:42_3"><span class="lab"><font color="Green" title="E58">A79</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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> )</span><br/><a NAME="E4:42_3_7_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,2)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c12">rf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_3_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c31">pp2</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_3_7"><span class="lab"><font color="Green" title="E64">A85</font></span></a>, <a class="txt" href="#E24:42_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="txt" href="#E30:42_3"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="txt" href="#E26:42_3"><span class="lab"><font color="Green" title="E59">A80</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"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E6:42_3_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_3_7_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:42_3_7_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_0.html#K7" title="MATRIX_0:func.7">Line</a> (<font color="Maroon" title="c14">N</font>,3)</span>)</span> <a href="fvsum_1.html#K13" title="FVSUM_1:func.13">"*"</a> <font color="Maroon" title="c12">rf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="anproj_9.html#E17"><span class="lab"><font color="Green" title="E16">Lem05</font></span></a></span>;<br/>
<a NAME="E2:42_3_7_1_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></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="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c29">p331</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:42_3"><span class="lab"><font color="Green" title="E57">A78</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</a>, <a class="txt" href="#E27:42_3"><span class="lab"><font color="Green" title="E60">A81</font></span></a>, <a class="txt" href="#E30:42_3"><span class="lab"><font color="Green" title="E61">A82</font></span></a>, <a class="txt" href="#E11:42_3"><span class="lab"><font color="Green" title="E50">A71</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" onmouseover="rs('xcmplx_1/T87')" onmouseout="rh()">XCMPLX_1:87</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42_3_7_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font></span>)</span> <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c32">pp3</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_3_7"><span class="lab"><font color="Green" title="E65">A86</font></span></a>, <a class="ref" href="finseq_1.html#T40" onmouseover="rs('finseq_1/T40')" onmouseout="rh()">FINSEQ_1:40</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></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:42_3_7"/>
<font color="Maroon" title="c19">a</font> <a href="anproj_8.html#K3" title="ANPROJ_8:func.3">*</a> <font color="Maroon" title="c18">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">N</font> <a href="laplace.html#K11" title="LAPLACE:func.11">*</a> <font color="Maroon" title="c12">rf</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_3_7"><span class="lab"><font color="Green" title="E66">A87</font></span></a>, <a class="ref" href="finseq_1.html#T45" onmouseover="rs('finseq_1/T45')" onmouseout="rh()">FINSEQ_1:45</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="E32:42_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T50" onmouseover="rs('euclid_8/T50')" onmouseout="rh()">EUCLID_8:50</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E63">A88</font></span>:  <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a>  = 
 <a href="anproj_8.html#K2" title="ANPROJ_8:func.2">M2F</a> <span class="p1">(<span class="default"><a href="anproj_8.html#K1" title="ANPROJ_8:func.1">F2M</a> <a href="euclid_8.html#K4" title="EUCLID_8:func.4">&lt;e3&gt;</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_8.html#T86" onmouseover="rs('anproj_8/T86')" onmouseout="rh()">ANPROJ_8:86</a></span>
<br/>.= 
<font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv3</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_3"><span class="lab"><font color="Green" title="E47">A68</font></span></a>, <a class="txt" href="#E31:42_3"><span class="lab"><font color="Green" title="E62">A83</font></span></a>, <a class="txt" href="#E17:42"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E19:42_3"><span class="lab"><font color="Green" title="E55">A76</font></span></a>, <a class="txt" href="#E5:42_3"><span class="lab"><font color="Green" title="E46">A67</font></span></a>, <a class="ref" href="anproj_8.html#T83" onmouseover="rs('anproj_8/T83')" onmouseout="rh()">ANPROJ_8:83</a></span>
;<br/>
<a NAME="E34:42_3"/>
(  not <font color="Maroon" title="c16">nv3</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  not <font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv3</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  &amp;  <a href="anproj_1.html#R1" title="ANPROJ_1:pred.1">are_Prop</a> <font color="Maroon" title="c16">nv3</font>,<font color="Maroon" title="c19">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">nv3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_3"><span class="lab"><font color="Green" title="E48">A69</font></span></a>, <a class="txt" href="#E11:42_3"><span class="lab"><font color="Green" title="E50">A71</font></span></a>, <a class="ref" href="anproj_9.html#T3" target="_self" onmouseover="rs('anproj_9/T3')" onmouseout="rh()">Th04</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="E35:42_3"/>
<span class="p1">(<span class="default"><a href="anproj_8.html#K4" title="ANPROJ_8:func.4">homography</a> <font color="Maroon" title="c14">N</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">P3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="anproj_9.html#K7" title="ANPROJ_9:func.7">Dir001</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:42_3"><span class="lab"><font color="Green" title="E63">A88</font></span></a>, <a class="ref" href="euclid_8.html#D3" onmouseover="rs('euclid_8/D3')" onmouseout="rh()">EUCLID_8:def 3</a>, <a class="ref" href="anproj_1.html#T22" onmouseover="rs('anproj_1/T22')" onmouseout="rh()">ANPROJ_1:22</a>, <a class="txt" href="#E8:42_3"><span class="lab"><font color="Green" title="E49">A70</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>


</div>
