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<span class="kw">let </span><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p2</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> 3</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="rvsum_1.html#K23" title="RVSUM_1:func.23">|(</a><span class="default"><font color="Maroon" title="c1">p1</font>,<font color="Maroon" title="c2">p2</font></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">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">f1</font> = <font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c4">f2</font> = <font color="Maroon" title="c2">p2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T27" title="EUCLID:th.27">EUCLID:27</a></span>;<br/>
<span class="lab"><font color="Green" title="E18">A1</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">f1</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T27" target="_self" title="EUCLID_5:th.27">Th27</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
;<br/>
<span class="lab"><font color="Green" title="E19">A2</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f2</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_5.html#T27" target="_self" title="EUCLID_5:th.27">Th27</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
;<br/>
<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="c2">p2</font></span><a href="rvsum_1.html#K23" title="RVSUM_1:func.23">)|</a></span> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#D17" title="RVSUM_1:def.17">RVSUM_1:def 17</a></span>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="euclid_5.html#T28" target="_self" title="EUCLID_5:th.28">Th28</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T108" title="RVSUM_1:th.108">RVSUM_1:108</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:30"/>
<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="c2">p2</font></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">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K1" title="EUCLID_5:func.1">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K2" title="EUCLID_5:func.2">`2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p1</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p2</font> <a href="euclid_5.html#K3" title="EUCLID_5:func.3">`3</a> </span>)</span></span>)</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/>


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