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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">q1</font>, <font color="Maroon" title="c4">q2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 3; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">q1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">q2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span><br/>







<a NAME="E1:94"/><span class="lab"><font color="Green" title="E42">A1</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="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</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> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</font></span>)</span></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</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_8.html#T75" target="_self" title="EUCLID_8:th.75">Th75</a></span>;<br/>
<a NAME="E2:94"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</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> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">q2</font></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_8.html#T75" target="_self" title="EUCLID_8:th.75">Th75</a></span>;<br/>
<a NAME="E3:94"/>
<span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</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> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">q1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">q2</font></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_8.html#T75" target="_self" title="EUCLID_8:th.75">Th75</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:94"/>
<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="c1">p1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c2">p2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">q1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c4">q2</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c3">q1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c4">q2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c3">q1</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c2">p2</font>,<font color="Maroon" title="c4">q2</font></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="#E1:94"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="txt" href="#E2:94"><span class="lab"><font color="Green" title="E43">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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