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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>) <b>-&gt; </b>   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> = <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K7">+</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="binop_1.html#NM2">BinOp</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E2:28_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ,   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>) <b>-&gt; </b>   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> = <font color="Maroon">a<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<span class="p2">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E4:28_1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">r</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">r</font>,<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="binop_1.html#S4">BINOP_1:sch 4</a>();<br/></i>
<a NAME="E5:28_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <br/> for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  <font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Olive">r</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Olive">x</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">RLSMS</font> =  <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">w</font>,<font color="Maroon">v</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><a NAME="E2:28_1_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">u1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">v1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  &amp; <font color="Maroon">w1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  )
 <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<font color="Maroon">v</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">v1</font></span>)</span>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">v1</font></span>)</span> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u1</font> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T32">RVSUM_1:32</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">u</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><a NAME="E2:28_1_1_3"/><i><font color="Green">E60</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #) <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0
 <b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>;<br/><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #)</span>)</span> = 
<font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> the <a href="struct_0.html#U2">Zero</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #)
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,the <a href="struct_0.html#U2">Zero</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #)
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T33">RVSUM_1:33</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">w</font> =  <a href="rvsum_1.html#K6">-</a> <font color="Maroon">v1</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #) <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>take </b><font color="Maroon">w</font> = <font color="Maroon">w</font>;<br/><b>thus </b><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font> = 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="rvsum_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6">-</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v1</font> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">v1</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52">RVSUM_1:52</a></i>
<br/>.= 
<font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> 0
<b>by </b><i><a class="ref" href="rvsum_1.html#T58">RVSUM_1:58</a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #)
<b>by </b><i><a class="ref" href="rlvect_1.html#D2">RLVECT_1:def 2</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5"><b>now </b></a><div class="add"><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font>, <font color="Maroon">w2</font> = <font color="Maroon">w1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>thus </b><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">w</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="rvsum_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">w2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73">RVSUM_1:73</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">w</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">w</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T72">RVSUM_1:72</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">v</font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span>,<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="real_1.html#K4">*</a> <font color="Maroon">b</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">v</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><b>thus </b>1 <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> 1,<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
1 <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rvsum_1.html#T74">RVSUM_1:74</a></i>
;<br/></div>
<b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E64</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">b</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">c</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">a1</font> = <font color="Maroon">a</font>, <font color="Maroon">b1</font> = <font color="Maroon">b</font>, <font color="Maroon">c1</font> = <font color="Maroon">c</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>thus </b><a NAME="E2:28_1_1_6"/>
(  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_1_6_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:28_1_1_6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E3:28_1_1_6_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">b1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:28_1_1_6_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="euclid.html#T19">EUCLID:19</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:28_1_1_6"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6_2"><b>proof </b></a><div class="add">

<a NAME="E1:28_1_1_6_2"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#T19">EUCLID:19</a></i>;<br/>
<a NAME="E2:28_1_1_6_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<b>hence </b><a NAME="E3:28_1_1_6_2"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b> <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">b1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">a1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T21">EUCLID:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
<br/>.= 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/><i><font color="Green">E65</font></i>:  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">b1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
;<br/><i><font color="Green">E66</font></i>:  <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">c1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
;<br/> <a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">c</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">b1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">c1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
;<br/><b>hence </b><a NAME="E8:28_1_1_6"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="euclid.html#T22">EUCLID:22</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E70</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>reconsider </b><font color="Maroon">v2</font> = <font color="Maroon">v1</font>, <font color="Maroon">w2</font> = <font color="Maroon">w1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><i><font color="Green">E72</font></i>:  <a href="metric_1.html#K2">dist</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
;<br/><i><font color="Green">E76</font></i>: <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> = 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">v1</font>
<b>by </b><i/>
;<br/><i><font color="Green">E77</font></i>: <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> = 
<font color="Maroon">g</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D4">RLVECT_1:def 4</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">w1</font>
<b>by </b><i/>
;<br/><b>thus </b> <a href="metric_1.html#K2">dist</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i>, , <a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K6">-</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T71">RVSUM_1:71</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <font color="Maroon">v2</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K6">-</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T76">RVSUM_1:76</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rvsum_1.html#K10">*</a> <span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="rvsum_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K6">-</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T73">RVSUM_1:73</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">w1</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T52">RVSUM_1:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">v</font>,<font color="Maroon">w</font></span>)</span>
<b>by </b><i>, <a class="ref" href="euclid.html#T14">EUCLID:14</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E80</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #), <font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #);<br/><b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font>, <font color="Maroon">w1</font> = <font color="Maroon">w</font>, <font color="Maroon">v1</font> = <font color="Maroon">v</font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/><b>reconsider </b><font color="Maroon">u2</font> = <font color="Maroon">u1</font>, <font color="Maroon">w2</font> = <font color="Maroon">w1</font>, <font color="Maroon">v2</font> = <font color="Maroon">v1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">n</font> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="euclid.html#D1">EUCLID:def 1</a></i>;<br/><i><font color="Green">E81</font></i>: <font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> = 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">u1</font>
<b>by </b><i/>
;<br/><i><font color="Green">E82</font></i>: <font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font> = 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">w</font>,<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D3">RLVECT_1:def 3</a></i>
<br/>.= 
<font color="Maroon">w1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">u1</font>
<b>by </b><i/>
;<br/><b>thus </b> <a href="metric_1.html#K2">dist</a> <font color="Maroon">v</font>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">v</font>,<font color="Maroon">w</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">w1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">v2</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">u2</font></span>)</span> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">u2</font></span>)</span> <a href="rvsum_1.html#K8">-</a> <font color="Maroon">w2</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T63">RVSUM_1:63</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">v2</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">u2</font></span>)</span> <a href="rvsum_1.html#K8">-</a> <span class="p3">(<span class="default"><font color="Maroon">u2</font> <a href="rvsum_1.html#K4">+</a> <font color="Maroon">w2</font></span>)</span></span>)</span></span><a href="euclid.html#K12">.|</a></span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T60">RVSUM_1:60</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">w1</font> <a href="euclid.html#K7">+</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>
<br/>.= 
 <a href="metric_1.html#K2">dist</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K2">+</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">RLSMS</font> =  <a href="vectmetr.html#G1" target="_self">RLSMetrStruct</a>(# <span class="p1">(<span class="default"><a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">f</font>,<font color="Maroon">g</font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="rlvect_1.html#V7">RealLinearSpace-like</a> <a href="vectmetr.html#V4" target="_self">strict</a>  <a href="vectmetr.html#L1" target="_self">RLSMetrStruct</a>  <b>by </b><i>, , , , , <a class="ref" href="rlvect_1.html#D5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8">RLVECT_1:def 8</a>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">RLSMS</font> = <font color="Maroon">RLSMS</font> as  <a href="vectmetr.html#V4" target="_self">strict</a> <a href="vectmetr.html#NM2">RealLinearMetrSpace</a> <b>by </b><i>, , , , , <a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a>, <a class="ref" href="metric_1.html#T3">METRIC_1:3</a>, <a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<b>take </b>
<font color="Maroon">RLSMS</font>
;<br/>

<b>thus </b><a NAME="E16:28_1_1"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RLSMS</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; the <a href="metric_1.html#U1">distance</a> of <font color="Maroon">RLSMS</font> <a href="funct_2.html#R1">=</a>  <a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font> &amp; the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">RLSMS</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">RLSMS</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="euclid.html#K7">+</a> <font color="Olive">y</font> ) &amp; ( for <font color="Olive">x</font> being   <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><br/> for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">RLSMS</font> <a href="binop_1.html#K1">.</a> <font color="Olive">r</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Olive">x</font> ) )
 <b>by </b><i>, </i>;<br/>


</div>
