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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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> <font color="Maroon" title="c1">n</font>,   <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> <font color="Maroon" title="c1">n</font>) <b>-&gt; </b>   <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> <font color="Maroon" title="c1">n</font> = $1 <a href="euclid.html#K7" title="EUCLID:func.7">+</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E2:28_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <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 href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>) <b>-&gt; </b>   <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> <font color="Maroon" title="c1">n</font> = $1 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> $2;<br/>
<b>consider </b><font color="Maroon" title="c3">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<span class="p2">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:28_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
for <font color="Olive" title="b1">r</font> being   <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> <br/> for <font color="Olive" title="b2">x</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> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">x</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></i>
<b>set </b><font color="Maroon" title="c4">RLSMS</font> =  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #);<br/>
<div><i><font color="Green" title="E13">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font></span><br/><b>reconsider </b><font color="Maroon" title="c7">v1</font> = <font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c8">w1</font> = <font color="Maroon" title="c6">w</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>thus </b><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> = 
<font color="Maroon" title="c7">v1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c8">w1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E14">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c8">u1</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c9">v1</font> = <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c10">w1</font> = <font color="Maroon" title="c7">w</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font> = 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">u1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c9">v1</font></span>)</span>,<font color="Maroon" title="c7">w</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">u1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c9">v1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c10">w1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c8">u1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c10">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="finseqop.html#T29" title="FINSEQOP:th.29">FINSEQOP:29</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">u</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c10">w1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E15">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font></span><br/><b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c5">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/><b>thus </b><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #)</span>)</span> = 
<font color="Maroon" title="c6">v1</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T33" title="RVSUM_1:th.33">RVSUM_1:33</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E8:28_1_1"/><i><font color="Green" title="E16">A6</font></i>: 
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) is <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">v</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c5">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">w</font> =  <a href="rvsum_1.html#K7" title="RVSUM_1:func.7">-</a> <font color="Maroon" title="c6">v1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) ;<br/>
<b>take </b>
<font color="Maroon" title="c7">w</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #)</span><br/>

<b>thus </b><font color="Maroon" title="c5">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font> = 
<font color="Maroon" title="c6">v1</font> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c6">v1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #)
<b>by </b><i><a class="ref" href="rvsum_1.html#T58" title="RVSUM_1:th.58">RVSUM_1:58</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E17">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5"><b>now </b></a><div class="add"><div><b>hereby </b> <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> ) )</span>
<div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">v</font>, <font color="Olive" title="b2">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">w</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c7">w</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c10">v2</font> = <font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c11">w2</font> = <font color="Maroon" title="c9">w1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/><b>thus </b><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">w</font></span>)</span> = 
<font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c9">w1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v2</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c11">w2</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">w1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">w1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><div><b>hereby </b> <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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b3">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> ) )</span>
<div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c7">v</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">v2</font> = <font color="Maroon" title="c8">v1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c9">v2</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72" title="RVSUM_1:th.72">RVSUM_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><div><b>hereby </b> <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">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c7">v</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">v2</font> = <font color="Maroon" title="c8">v1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c9">v2</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>
<br/>.= 
<font color="Maroon" title="c3">g</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">a</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span>
<div class="add"><b>let </b><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v</font></span><br/><b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c5">v</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/><b>thus </b>1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> = 
1 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c6">v1</font>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">v</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74" title="RVSUM_1:th.74">RVSUM_1:74</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E18">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> ) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span> )</span><br/><b>reconsider </b><font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c9">b1</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c10">c1</font> = <font color="Maroon" title="c7">c</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>thus </b><a NAME="E2:28_1_1_6"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_1_6_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font></span><br/>

<a NAME="E2:28_1_1_6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E3:28_1_1_6_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">b1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_6_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="euclid.html#T19" title="EUCLID:th.19">EUCLID:19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:28_1_1_6"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_1_1_6_2"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">a1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#T19" title="EUCLID:th.19">EUCLID:19</a></i>;<br/>
<a NAME="E2:28_1_1_6_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:28_1_1_6_2"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>thus </b> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">b1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">a1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T21" title="EUCLID:th.21">EUCLID:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
<br/>.= 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span></span><br/><i><font color="Green" title="E19">A9</font></i>:  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">b1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
;<br/><i><font color="Green" title="E20">A10</font></i>:  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">c1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
;<br/> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c10">c1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
;<br/><b>hence </b><a NAME="E8:28_1_1_6"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c7">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font></span>)</span>
 <b>by </b><i><a class="txt" href="vectmetr.html#E5:28_1_1_6"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="vectmetr.html#E6:28_1_1_6"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="euclid.html#T22" title="EUCLID:th.22">EUCLID:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E19">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">v</font>, <font color="Olive" title="b2">w</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c7">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c8">v1</font> = <font color="Maroon" title="c6">v</font>, <font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c7">w</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c10">v2</font> = <font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c11">w2</font> = <font color="Maroon" title="c9">w1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/><i><font color="Green" title="E20">A12</font></i>:  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">w1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
;<br/><a NAME="E4:28_1_1_7"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c8">v1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/><a NAME="E5:28_1_1_7"/><i><font color="Green" title="E22">A14</font></i>: 
<font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c9">w1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>;<br/><b>thus </b> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w</font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c10">v2</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="txt" href="vectmetr.html#E4:28_1_1_7"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="vectmetr.html#E5:28_1_1_7"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c10">v2</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <font color="Maroon" title="c10">v2</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K7" title="RVSUM_1:func.7">-</a> <font color="Maroon" title="c11">w2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71" title="RVSUM_1:th.71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">v1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">w1</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73" title="RVSUM_1:th.73">RVSUM_1:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c7">w</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E3:28_1_1_7"><i><font color="Green" title="E20">A12</font></i></a>, <a class="ref" href="euclid.html#T14" title="EUCLID:th.14">EUCLID:14</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E20">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">w</font>, <font color="Maroon" title="c7">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font></span>)</span></span><br/><b>reconsider </b><font color="Maroon" title="c8">u1</font> = <font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c9">w1</font> = <font color="Maroon" title="c6">w</font>, <font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c7">v</font> as    <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> <font color="Maroon" title="c1">n</font> ;<br/><b>reconsider </b><font color="Maroon" title="c11">u2</font> = <font color="Maroon" title="c8">u1</font>, <font color="Maroon" title="c12">w2</font> = <font color="Maroon" title="c9">w1</font>, <font color="Maroon" title="c13">v2</font> = <font color="Maroon" title="c10">v1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <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>  ;<br/><a NAME="E3:28_1_1_8"/><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c8">u1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><a NAME="E4:28_1_1_8"/><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c6">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">w1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c8">u1</font>
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a></i>;<br/><b>thus </b> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c6">w</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c6">w</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">w1</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c13">v2</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c11">u2</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c11">u2</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <font color="Maroon" title="c12">w2</font></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T63" title="RVSUM_1:th.63">RVSUM_1:63</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">v2</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c11">u2</font></span>)</span> <a href="rvsum_1.html#K9" title="RVSUM_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">u2</font> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <font color="Maroon" title="c12">w2</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T60" title="RVSUM_1:th.60">RVSUM_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c8">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">w1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c8">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D6" title="EUCLID:def.6">EUCLID:def 6</a></i>
<br/>.= 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">u</font></span>)</span>
<b>by </b><i><a class="txt" href="vectmetr.html#E3:28_1_1_8"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="vectmetr.html#E4:28_1_1_8"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c5">RLSMS</font> =  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">g</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">RealLinearSpace-like</a> <a href="vectmetr.html#V4" title="VECTMETR:attr.4">strict</a>  <a href="vectmetr.html#L1" title="VECTMETR:struct.1">RLSMetrStruct</a>  <b>by </b><i><a class="txt" href="vectmetr.html#E5:28_1_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="vectmetr.html#E6:28_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="vectmetr.html#E7:28_1_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="vectmetr.html#E8:28_1_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="vectmetr.html#E9:28_1_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">RLSMS</font> = <font color="Maroon" title="c5">RLSMS</font> as  <a href="vectmetr.html#V4" title="VECTMETR:attr.4">strict</a> <a href="vectmetr.html#NM2" title="VECTMETR:NM.2">RealLinearMetrSpace</a> <b>by </b><i><a class="txt" href="vectmetr.html#E10:28_1_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="vectmetr.html#E11:28_1_1"><i><font color="Green" title="E19">A11</font></i></a>, <a class="txt" href="vectmetr.html#E12:28_1_1"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="vectmetr.html#D5" target="_self" title="VECTMETR:def.5">Def5</a>, <a class="ref" href="vectmetr.html#D6" target="_self" title="VECTMETR:def.6">Def6</a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a>, <a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">RLSMS</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">RLSMS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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> <font color="Maroon" title="c1">n</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">y</font> ) &amp; ( for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">r</font> being   <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>  holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<b>thus </b><a NAME="E15:28_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c6">RLSMS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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> <font color="Maroon" title="c1">n</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">y</font> ) &amp; ( for <font color="Olive" title="b1">x</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> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">r</font> being   <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>  holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c6">RLSMS</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x</font> ) )
 <b>by </b><i><a class="txt" href="vectmetr.html#E2:28_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="vectmetr.html#E4:28_1_1"><i><font color="Green" title="E12">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
