<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">r1</font>, <font color="Maroon" title="c2">r2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <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 holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span></span><br/>





<a NAME="E1:86"/>
for <font color="Olive" title="b1">R</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b1">R</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Olive" title="b1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">R</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 3 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</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">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="binop_2.html#K7" title="BINOP_2:func.7">-</a> <font color="Maroon" title="c2">r2</font></span>)</span></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:86_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c4">R</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>
<a NAME="E2:86"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">p</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:86"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">r2</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">r1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c2">r2</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1"><a href="euclid_8.html#K1" title="EUCLID_8:func.1">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 3</span>)</span></span><a href="euclid_8.html#K1" title="EUCLID_8:func.1">]|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="euclid_8.html#T1" target="_self" title="EUCLID_8:th.1">Lm1</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>
