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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">E</font>, <font color="Maroon" title="c3">F</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</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"> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">F</font>)</span>)</span></span>)</span></span>)</span></span><br/>





<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">F</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T7" onmouseover="rs('euclid_6/T7')" onmouseout="rh()">EUCLID_6:7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">F</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T43" onmouseover="rs('euclid_6/T43')" onmouseout="rh()">EUCLID_6:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">E</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">F</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T43" onmouseover="rs('euclid_6/T43')" onmouseout="rh()">EUCLID_6:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">F</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T43" onmouseover="rs('euclid_6/T43')" onmouseout="rh()">EUCLID_6:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">F</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T43" onmouseover="rs('euclid_6/T43')" onmouseout="rh()">EUCLID_6:43</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:5"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K1" title="SQUARE_1:func.1">^2</a></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p4"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c2">E</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">F</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K19" title="SIN_COS:func.19">cos</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> (<font color="Maroon" title="c2">E</font>,<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">F</font>)</span>)</span></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"> verum</span><br/>


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