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<span class="kw">let </span><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</font>, <font color="Maroon" title="c3">p3</font>, <font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> implies ( (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; ( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:38"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> or  not <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> or ( (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; ( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:38"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font> or ( (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; ( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:38"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p3</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; ( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1"/>
( <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; ( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) )</span><br/><div class="add"><a NAME="E2:38_1_1"/><span class="kw">then </span>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rltopsp1.html#T71" title="RLTOPSP1:th.71">RLTOPSP1:71</a></span>;<br/><a NAME="E3:38_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:38_1_1"/>
<font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_1_1"/>
( (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; ( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="ref" href="euclid_2.html#T54" title="EUCLID_2:th.54">EUCLID_2:54</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2"/>
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> ) &amp; ( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:38_1_2"/>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> implies <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1_2_1"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span><br/>

<span class="lab"><font color="Green" title="E40">A7</font></span>: <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T7" target="_self" title="EUCLID_6:th.7">Th7</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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E1:38_1_2_1"><span class="lab"><font color="Green" title="E39">A6</font></span></a></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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T7" target="_self" title="EUCLID_6:th.7">Th7</a></span>
;<br/>
<span class="kw">thus </span><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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="euclid_6.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>
<br/>.= 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_2_1"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="euclid_6.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:38_1_2"/>
( <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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> implies <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1_2_2"/><span class="lab"><font color="Green" title="E39">A8</font></span>: 
<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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_2_2_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font> or <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_2_1_1"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:38_1_2_2_1_1"/><span class="kw">then </span>
<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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="euclid_6.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2_2_1_1"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="txt" href="euclid_6.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T46" title="EUCLID:th.46">EUCLID:46</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_2.html#T54" title="EUCLID_2:th.54">EUCLID_2:54</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_2_1_2"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><div class="add"><a NAME="E2:38_1_2_2_1_2"/><span class="kw">then </span>
<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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="euclid_6.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E3:38_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="euclid_6.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E4:38_1_2_2_1_2"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
 <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_2_2_1_2_1_1"/>
( <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  or <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E1:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E3:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="euclid_6.html#T13" target="_self" title="EUCLID_6:th.13">Th13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span> <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> = 
 <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T84" title="SIN_COS:th.84">SIN_COS:84</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T34" title="SIN_COS:th.34">SIN_COS:34</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">hence </span> <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span> = 
 <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T84" title="SIN_COS:th.84">SIN_COS:84</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T84" title="SIN_COS:th.84">SIN_COS:84</a></span>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="sin_cos.html#T34" title="SIN_COS:th.34">SIN_COS:34</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E5:38_1_2_2_1_2"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
( <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">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^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">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span></span>)</span></span>)</span> &amp; <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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^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="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_6.html#T7" target="_self" title="EUCLID_6:th.7">Th7</a></span>;<br/><a NAME="E6:38_1_2_2_1_2"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<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="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2_2"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="euclid_6.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><a NAME="E7:38_1_2_2_1_2"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="euclid_6.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E8:38_1_2_2_1_2"/><span class="lab"><font color="Green" title="E46">A16</font></span>: 
<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">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="euclid_6.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E9:38_1_2_2_1_2"/>
<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">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">p3</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="euclid_6.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><a NAME="E10:38_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E6:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E44">A14</font></span></a></span>;<br/><a NAME="E11:38_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">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="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <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="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E6:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E12:38_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A17</font></span>: 
2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="sin_cos.html#K25" title="SIN_COS:func.25">cos</a> <span class="p2">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E45">A15</font></span></a>, <a class="ref" href="xcmplx_1.html#T5" title="XCMPLX_1:th.5">XCMPLX_1:5</a></span>;<br/><a NAME="E13:38_1_2_2_1_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font> &amp;  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex2.html#T84" title="COMPLEX2:th.84">COMPLEX2:84</a></span>;<br/><a NAME="E14:38_1_2_2_1_2"/><span class="kw">then </span>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E47">A17</font></span></a>, <a class="txt" href="#E4:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E42">A12</font></span></a>, <a class="ref" href="comptrig.html#T34" title="COMPTRIG:th.34">COMPTRIG:34</a></span>;<br/><a NAME="E15:38_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="euclid_3.html#T54" title="EUCLID_3:th.54">EUCLID_3:54</a></span>;<br/><span class="kw">hence </span><a NAME="E16:38_1_2_2_1_2"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E1:38_1_2_2_1_2"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="euclid_6.html#T17" target="_self" title="EUCLID_6:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:38_1_2"/>
( <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:38_1_2_3"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E40">A19</font></span>:  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>

<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c1">p1</font></span>)</span></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_2.html#T44" title="EUCLID_2:th.44">EUCLID_2:44</a></span>
<br/>.= 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c2">p2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T48" title="EUCLID:th.48">EUCLID:48</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_3_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_2_3_2"/>
( ( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> ) or <font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> or <font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_3_2_1"/>
( <font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp; <font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:38_1_2_3_2_1"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_3_2_2"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><a NAME="E2:38_1_2_3_2_2"/><span class="kw">then </span>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E2:38_1_2_3"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="euclid_6.html#T17" target="_self" title="EUCLID_6:th.17">Th17</a></span>;<br/><a NAME="E3:38_1_2_3_2_2"/><span class="kw">then </span>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:38_1_2_3_2_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="euclid_3.html#T54" title="EUCLID_3:th.54">EUCLID_3:54</a></span>;<br/><span class="kw">hence </span><a NAME="E4:38_1_2_3_2_2"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:38_1_2_3_2_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="euclid_6.html#T18" target="_self" title="EUCLID_6:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_3_2_3"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_3_2_3"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
<font color="Maroon" title="c2">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><a NAME="E2:38_1_2_3_2_3"/><span class="kw">then </span>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p3</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E1:38_1_2_3"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="euclid_6.html#T17" target="_self" title="EUCLID_6:th.17">Th17</a></span>;<br/><a NAME="E3:38_1_2_3_2_3"/><span class="kw">then </span>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2 or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">3 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:38_1_2_3_2_3"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="euclid_3.html#T54" title="EUCLID_3:th.54">EUCLID_3:54</a></span>;<br/><a NAME="E4:38_1_2_3_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A22</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c1">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:38"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:38_1_2_3_2_3"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="euclid_6.html#T18" target="_self" title="EUCLID_6:th.18">Th18</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_3_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_2_3_2_3_1"/>
(  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_3_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_3_2_3_1_1"/><span class="lab"><font color="Green" title="E43">A23</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><a NAME="E2:38_1_2_3_2_3_1_1"/><span class="kw">then </span>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_3.html#T45" title="EUCLID_3:th.45">EUCLID_3:45</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_2_3_2_3_1_1"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_2_3_2_3"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E1:38_1_2_3_2_3_1_1"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="euclid_3.html#T45" title="EUCLID_3:th.45">EUCLID_3:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_2_3_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_2_3_2_3_1_2"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font></span><br/><div class="add"><a NAME="E2:38_1_2_3_2_3_1_2"/><span class="kw">then </span>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <a href="sin_cos.html#K36" title="SIN_COS:func.36">PI</a> </span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_3.html#T46" title="EUCLID_3:th.46">EUCLID_3:46</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_2_3_2_3_1_2"/>
 <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid_3.html#K4" title="EUCLID_3:func.4">angle</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c3">p3</font>,<font color="Maroon" title="c2">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_2_3_2_3"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E1:38_1_2_3_2_3_1_2"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="euclid_3.html#T46" title="EUCLID_3:th.46">EUCLID_3:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
