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<b>let </b><font color="Maroon" title="c1">x1</font>, <font color="Maroon" title="c2">x2</font>, <font color="Maroon" title="c3">y1</font>, <font color="Maroon" title="c4">y2</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E23">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">y2</font> )
 ;<br/>

<a NAME="E2:27"/><b>then </b>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T19" target="_self" title="EUCLID_2:th.19">Th19</a></i>;<br/>
<a NAME="E3:27"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c3">y1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">y1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">y1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="euclid_2.html#T19" target="_self" title="EUCLID_2:th.19">Th19</a></i>;<br/>
<a NAME="E4:27"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x1</font>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="euclid_2.html#T6" target="_self" title="EUCLID_2:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E5:27"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c2">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y1</font> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <font color="Maroon" title="c4">y2</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</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="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c3">y1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x1</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c3">y1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x2</font>,<font color="Maroon" title="c4">y2</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
 <b>by </b><i><a class="txt" href="euclid_2.html#E1:27"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="euclid_2.html#E3:27"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="euclid_2.html#T19" target="_self" title="EUCLID_2:th.19">Th19</a></i>;<br/>


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