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

<span class="kw">reconsider </span><font color="Maroon" title="c1">ZS</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">O</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">FF</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"><font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c1">ZS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw"> such that </span><br/><a NAME="E4:23_1_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> holds  <font color="Maroon" title="c3">FF</font> <a href="metric_1.html#K1" title="METRIC_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="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> = <font color="Maroon" title="c2">O</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> = <font color="Maroon" title="c2">O</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">F</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c1">ZS</font><span class="kw"> such that </span><br/><a NAME="E6:23_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> holds  <font color="Maroon" title="c4">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>3</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></span>
<span class="kw">consider </span><font color="Maroon" title="c5">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> ,<font color="Maroon" title="c1">ZS</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">ZS</font><span class="kw"> such that </span><br/><a NAME="E8:23_1_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
for <font color="Olive" title="b1">a</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> holds  <font color="Maroon" title="c5">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</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">a</font>,<font color="Olive" title="b2">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S4" title="BINOP_1:sch.4">BINOP_1:sch 4</a>();<br/></span>
<span class="kw">set </span><font color="Maroon" title="c6">W</font> =  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #);<br/>
<a NAME="E9:23_1_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b3">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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">x</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">x</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">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</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="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</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="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c9">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">b</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">X</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E3:23_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">a</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c13">X</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</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="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">take </span>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #)
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V4" title="VECTMETR:attr.4">strict</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E10:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V4" title="VECTMETR:attr.4">strict</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<a NAME="E11:23_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">Y</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E2:23_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c9">X</font>,<font color="Maroon" title="c10">Y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<a NAME="E13:23_1_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c9">z</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c8">y</font>, <font color="Maroon" title="c12">Z</font> = <font color="Maroon" title="c9">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E2:23_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c10">X</font>,<font color="Maroon" title="c11">Y</font>),<font color="Maroon" title="c12">Z</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c9">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<a NAME="E15:23_1_1"/>
for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c7">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E2:23_1_1_4"/>
<font color="Maroon" title="c7">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c2">O</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:23_1_1_4"/>
<font color="Maroon" title="c7">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E17:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c2">O</font> as   <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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) ;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">y</font>
; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #)</span><br/>

<span class="kw">thus </span><a NAME="E2:23_1_1_5"/>
<font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E18:23_1_1"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">y</font></span>)</span> <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="b2">x</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="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <font color="Maroon" title="c7">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">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</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="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">a</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x</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="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c11">X</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c10">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E3:23_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x</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="c8">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c11">X</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c12">Y</font>))
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:23_1_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_1_1_6"/>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x</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="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E19:23_1_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/> for <font color="Olive" title="b3">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">x</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">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c8">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c9">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">X</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E3:23_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c9">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c13">X</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:23_1_1"/>
for <font color="Olive" title="b1">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">A9</font> = 1 as    <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/>
<span class="kw">let </span><font color="Maroon" title="c8">x</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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">X</font> = <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E3:23_1_1_8"/>
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c7">A9</font>,<font color="Maroon" title="c9">X</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:23_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_1_1_8"/>
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E21:23_1_1"/>
(  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:23_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E9:23_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E19:23_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a>, <a class="ref" href="rlvect_1.html#D9" title="RLVECT_1:def.9">RLVECT_1:def 9</a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<a NAME="E22:23_1_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds <br/> (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">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="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">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="c7">a</font>,<font color="Maroon" title="c8">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="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> ) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c7">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span> )</span><br/>

<span class="lab"><font color="Green" title="E18">A8</font></span>:  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font> = 
<font color="Maroon" title="c3">FF</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:23_1_1_9"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c7">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> ) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c7">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span> )</span><br/>

<a NAME="E3:23_1_1_9"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:23_1_1_9"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c7">a</font> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span> )</span><br/>

<a NAME="E5:23_1_1_9"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:23_1_1_9"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1_9"><span class="lab"><font color="Green" title="E19">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E7:23_1_1_9"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c9">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="c7">a</font>,<font color="Maroon" title="c8">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="c8">b</font>,<font color="Maroon" title="c9">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:23_1_1_9"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E5:23_1_1_9"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E1:23_1_1_9"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E23:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E24:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:23_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E25:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:23_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="metric_1.html#T3" title="METRIC_1:th.3">METRIC_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E26:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:23_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a>  &amp;  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a>  )</span><br/>

<a NAME="E27:23_1_1"/>
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">v</font>, <font color="Olive" title="b3">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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">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="Olive" title="b1">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="b2">v</font>,<font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">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="c7">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="c7">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="c7">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/><span class="kw">let </span><font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c9">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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">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="c7">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="c8">v</font>,<font color="Maroon" title="c9">w</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c10">v1</font> = <font color="Maroon" title="c8">v</font>, <font color="Maroon" title="c11">w1</font> = <font color="Maroon" title="c9">w</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">ZS</font> ;<br/>
<a NAME="E2:23_1_1_10"/><span class="lab"><font color="Green" title="E18">X</font></span>: 
<font color="Maroon" title="c3">FF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">w1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>;<br/>
<span class="kw">thus </span> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">w</font></span>)</span> = 
<font color="Maroon" title="c3">FF</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">w1</font>)

<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">FF</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c10">v1</font>,<font color="Maroon" title="c11">w1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:23_1_1_10"><span class="lab"><font color="Green" title="E18">X</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c7">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="c8">v</font>,<font color="Maroon" title="c9">w</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E28:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V5" title="VECTMETR:attr.5">homogeneous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#D5" target="_self" title="VECTMETR:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a> </span><br/>

<a NAME="E29:23_1_1"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">w</font>, <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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b2">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="Olive" title="b3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">u</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">u</font>, <font color="Maroon" title="c8">w</font>, <font color="Maroon" title="c9">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>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">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="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">u</font></span>)</span></span><br/>

<span class="kw">thus </span> <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">u</font></span>)</span> = 
<font color="Maroon" title="c3">FF</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">FF</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:23_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a></span>
<br/>.= 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E30:23_1_1"/>
 <a href="vectmetr.html#G1" title="VECTMETR:aggr.1">RLSMetrStruct</a>(# <font color="Maroon" title="c1">ZS</font>,<font color="Maroon" title="c3">FF</font>,<font color="Maroon" title="c2">O</font>,<font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">G</font> #) is  <a href="vectmetr.html#V6" title="VECTMETR:attr.6">translatible</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectmetr.html#D6" target="_self" title="VECTMETR:def.6">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
