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

<span class="kw">let </span><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">V2</font> be   <a href="vectmetr.html#V4" title="VECTMETR:attr.4">strict</a>  <a href="vectmetr.html#NM2" title="VECTMETR:NM.2">RealLinearMetrSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">V1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">y</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x</font> ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> &amp; the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c3">V2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">y</font> ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28_1_2"/><span class="lab"><font color="Green" title="E11">A16</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E2:28_1_2"/><span class="lab"><font color="Green" title="E12">A17</font></span>: 
( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c2">V1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">and </span><br/><a NAME="E3:28_1_2"/><span class="lab"><font color="Green" title="E13">A18</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">y</font>
 <span class="kw">and </span><br/><a NAME="E4:28_1_2"/><span class="lab"><font color="Green" title="E14">A19</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x</font>
 <span class="kw">and </span><br/><a NAME="E5:28_1_2"/><span class="lab"><font color="Green" title="E15">A20</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E6:28_1_2"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
( the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c3">V2</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="euclid.html#K13" title="EUCLID:func.13">Pitag_dist</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">V2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">and </span><br/><a NAME="E7:28_1_2"/><span class="lab"><font color="Green" title="E17">A22</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> holds  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Olive" title="b2">y</font>
 <span class="kw">and </span><br/><a NAME="E8:28_1_2"/><span class="lab"><font color="Green" title="E18">A23</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font><br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_2"><span class="lab"><font color="Green" title="E11">A16</font></span></a></span>;<br/><span class="kw">thus </span>the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font> = 
<font color="Maroon" title="c6">x1</font> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <font color="Maroon" title="c7">y1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2"><span class="lab"><font color="Green" title="E13">A18</font></span></a></span>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_1_2"><span class="lab"><font color="Green" title="E17">A22</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A24</font></span>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c2">V1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">V2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_2"><span class="lab"><font color="Green" title="E11">A16</font></span></a>, <a class="txt" href="#E5:28_1_2"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</font> holds  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">r</font>,<font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">r</font>,<font color="Olive" title="b1">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">x</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c5">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_2"><span class="lab"><font color="Green" title="E11">A16</font></span></a></span>;<br/><span class="kw">thus </span>the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c2">V1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">x</font> = 
<font color="Maroon" title="c4">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c6">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1_2"><span class="lab"><font color="Green" title="E14">A19</font></span></a></span>
<br/>.= 
the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c3">V2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_1_2"><span class="lab"><font color="Green" title="E18">A23</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:28_1_2"/>
<font color="Maroon" title="c2">V1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">V2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_2"><span class="lab"><font color="Green" title="E11">A16</font></span></a>, <a class="txt" href="#E2:28_1_2"><span class="lab"><font color="Green" title="E12">A17</font></span></a>, <a class="txt" href="#E5:28_1_2"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="txt" href="#E6:28_1_2"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="txt" href="#E10:28_1_2"><span class="lab"><font color="Green" title="E19">A24</font></span></a>, <a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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