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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> being   <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>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">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="Olive" title="b2">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">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="Olive" title="b3">x3</font> holds <br/><span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Olive" title="b1">x1</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span>)</span>,<font color="Olive" title="b3">x3</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"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b3">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font>, <font color="Maroon" title="c5">x3</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c4">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="c5">x3</font> implies <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">x2</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x3</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"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green" title="E20">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">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="c4">x2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">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="c5">x3</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">x2</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x3</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"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span></span><br/>

<a NAME="E2:24"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">x1</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="c3">x1</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8" target="_self" title="EUCLID_2:th.8">Th8</a></i>;<br/>
<a NAME="E3:24"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">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="c4">x2</font>
 
<b>by </b><i><a class="ref" href="euclid_2.html#T8" target="_self" title="EUCLID_2:th.8">Th8</a></i>;<br/>
<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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">x2</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> = 
<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">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">x1</font></span>)</span>,<font color="Maroon" title="c5">x3</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="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<font color="Maroon" title="c5">x3</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#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="euclid_2.html#E2:24"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="euclid_2.html#T19" target="_self" title="EUCLID_2:th.19">Th19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">x2</font></span>)</span>,<font color="Maroon" title="c5">x3</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#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="euclid_2.html#T20" target="_self" title="EUCLID_2:th.20">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</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#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="euclid_2.html#T20" target="_self" title="EUCLID_2:th.20">Th20</a></i>
;<br/>
<b>hence </b><a NAME="E5:24"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c3">x1</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon" title="c4">x2</font></span>)</span></span>)</span>,<font color="Maroon" title="c5">x3</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"><font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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