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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><span class="kw">consider </span><font color="Maroon" title="c4">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c5">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E2:39_1_1_1"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="prvect_3.html#E7"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">v</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E4:39_1_1_1"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K6" title="PRVECT_3:func.6">prod_NORM</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" target="_self" title="PRVECT_3:def.6">Def6</a></span>;<br/><span class="kw">assume </span><a NAME="E5:39_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E6:39_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><a NAME="E7:39_1_1_1"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E8:39_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="finseq_2.html#T61" title="FINSEQ_2:th.61">FINSEQ_2:61</a></span>;<br/><span class="kw">hence </span><a NAME="E9:39_1_1_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <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="txt" href="#E4:39_1_1_1"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E2:39_1_1_1"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="euclid.html#T7" title="EUCLID:th.7">EUCLID:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E2:39_1_1"/><span class="kw">then </span>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:39_1_1"/>
<span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> is  <a href="normsp_0.html#V4" title="NORMSP_0:attr.4">reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D6" title="NORMSP_0:def.6">NORMSP_0:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp; <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a>  )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <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="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c4">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c5">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E2:39_1_1_2"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="prvect_3.html#E7"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c6">v</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E4:39_1_1_2"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K6" title="PRVECT_3:func.6">prod_NORM</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c6">v</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" target="_self" title="PRVECT_3:def.6">Def6</a></span>;<br/><span class="kw">assume </span><a NAME="E5:39_1_1_2"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span></span><br/><a NAME="E6:39_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_2"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E2:39_1_1_2"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="ref" href="euclid.html#T8" title="EUCLID:th.8">EUCLID:8</a></span>;<br/><a NAME="E7:39_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T61" title="FINSEQ_2:th.61">FINSEQ_2:61</a></span>;<br/><a NAME="E8:39_1_1_2"/>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_2"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E9:39_1_1_2"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c5">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <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="txt" href="#E7:39_1_1_2"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><a NAME="E10:39_1_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D5" title="NORMSP_0:def.5">NORMSP_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E11:39_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_2"><span class="lab"><font color="Green" title="E15">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:39_1_1"/>
<span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D5" title="NORMSP_0:def.5">NORMSP_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></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">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c7">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E2:39_1_1_3"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="prvect_3.html#E7"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c9">y2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c2">F</font><span class="kw"> such that </span><br/><a NAME="E4:39_1_1_3"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="prvect_3.html#E7"><span class="lab"><font color="Green" title="E6">Lm1</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">v</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E6:39_1_1_3"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K6" title="PRVECT_3:func.6">prod_NORM</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" target="_self" title="PRVECT_3:def.6">Def6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E8:39_1_1_3"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
( <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K6" title="PRVECT_3:func.6">prod_NORM</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c11">z</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" target="_self" title="PRVECT_3:def.6">Def6</a></span>;<br/><span class="kw">thus </span><a NAME="E9:39_1_1_3"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c12">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E2:39_1_1_3_1"/><span class="lab"><font color="Green" title="E19">A10</font></span>: 
( <font color="Maroon" title="c12">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K6" title="PRVECT_3:func.6">prod_NORM</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c12">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" target="_self" title="PRVECT_3:def.6">Def6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">aa</font> =  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c14">xx1</font> = <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>, <font color="Maroon" title="c15">xx2</font> = <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> as   <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/>
<a NAME="E4:39_1_1_3_1"/>
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">x1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x2</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c12">w</font> = 
<font color="Maroon" title="c13">aa</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><font color="Maroon" title="c14">xx1</font>,<font color="Maroon" title="c15">xx2</font></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_3_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="euclid.html#T58" title="EUCLID:th.58">EUCLID:58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c10">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_3"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="euclid.html#T65" title="EUCLID:th.65">EUCLID:65</a></span>
;<br/>
<span class="kw">then </span><span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c12">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T11" title="EUCLID:th.11">EUCLID:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>

;<br/>
<span class="kw">hence </span><a NAME="E7:39_1_1_3_1"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_3"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E2:39_1_1_3_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E2:39_1_1_3"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="prvect_3.html#D2" target="_self" title="PRVECT_3:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E10:39_1_1_3"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c12">w</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 2<span class="kw"> such that </span><br/><a NAME="E2:39_1_1_3_2"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
( <font color="Maroon" title="c12">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y2</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="prvect_3.html#K6" title="PRVECT_3:func.6">prod_NORM</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font>)</span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c12">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="prvect_3.html#D6" target="_self" title="PRVECT_3:def.6">Def6</a></span>;<br/>
<a NAME="E3:39_1_1_3_2"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c12">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_3_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E2:39_1_1_3"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E4:39_1_1_3"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="prvect_3.html#D1" target="_self" title="PRVECT_3:def.1">Def1</a></span>;<br/>
<a NAME="E4:39_1_1_3_2"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y2</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">t</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">t</font> = <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<a NAME="E6:39_1_1_3_2"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_3_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_3_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">w</font></span>)</span> implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:39_1_1_3_2_1"/>
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">w</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><a NAME="E2:39_1_1_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_3_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:39_1_1_3_2_1_1"/>
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_3_2_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_3_2_1_1_1"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:39_1_1_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_3_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_1_1_3_2_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_3_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E1:39_1_1_3_2_1_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a>, <a class="ref" href="normsp_1.html#T4" title="NORMSP_1:th.4">NORMSP_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_3_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_3_2_1_1_2"/><span class="lab"><font color="Green" title="E24">A17</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><a NAME="E2:39_1_1_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">y2</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1_3_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_1_1_3_2_1_1_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c12">w</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_3_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E1:39_1_1_3_2_1_1_2"><span class="lab"><font color="Green" title="E24">A17</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a>, <a class="ref" href="normsp_1.html#T4" title="NORMSP_1:th.4">NORMSP_1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E8:39_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A18</font></span>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c12">w</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c14">t</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_3_2"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="prvect_2.html#T2" title="PRVECT_2:th.2">PRVECT_2:2</a></span>;<br/>
<font color="Maroon" title="c14">t</font> = 
<span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c6">x1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c7">x2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1"><a href="euclid.html#K19" title="EUCLID:func.19">|[</a><span class="default"><span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c8">y1</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>,<span class="p2"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c9">y2</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span></span><a href="euclid.html#K19" title="EUCLID:func.19">]|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></span>
<br/>.= 
<font color="Maroon" title="c10">v</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c11">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39_1_1_3"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E8:39_1_1_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="euclid.html#T64" title="EUCLID:th.64">EUCLID:64</a></span>
;<br/>
<a NAME="E10:39_1_1_3_2"/><span class="kw">then </span>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c14">t</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c10">v</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c11">z</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#T12" title="EUCLID:th.12">EUCLID:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:39_1_1_3_2"/>
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1_3_2"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E2:39_1_1_3"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E6:39_1_1_3"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E4:39_1_1_3"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E8:39_1_1_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E8:39_1_1_3_2"><span class="lab"><font color="Green" title="E23">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:39_1_1"/>
<span class="p1"><a href="prvect_3.html#K7" title="PRVECT_3:func.7">[:</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">F</font></span><a href="prvect_3.html#K7" title="PRVECT_3:func.7">:]</a></span> is  <a href="normsp_1.html#V2" title="NORMSP_1:attr.2">RealNormSpace-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D1" title="NORMSP_1:def.1">NORMSP_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
