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

<span class="kw">set </span><font color="Maroon" title="c2">W</font> =  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #);<br/>
<a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
 for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">u9</font> = <font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c7">v9</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c8">w9</font> = <font color="Maroon" title="c5">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">v9</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">w9</font>

<br/>.= 
<font color="Maroon" title="c6">u9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">w9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:31_1_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <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">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">v9</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)</span>)</span> = 
<font color="Maroon" title="c4">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T4" title="RLVECT_1:th.4">RLVECT_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>
<a NAME="E3:31_1_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) is  <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">v</font> is  <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">v9</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">w9</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:31_1_1_3"/><span class="lab"><font color="Green" title="E26">A4</font></span>: 
<font color="Maroon" title="c4">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">w9</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">V</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">w</font> = <font color="Maroon" title="c5">w9</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">w</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)</span><br/>

<span class="kw">thus </span><a NAME="E5:31_1_1_3"/>
<font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_3"><span class="lab"><font color="Green" title="E26">A4</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>
<a NAME="E4:31_1_1"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> 1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">v9</font> = <font color="Maroon" title="c3">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">thus </span>1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">v</font> = 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v9</font>

<br/>.= 
<font color="Maroon" title="c3">v</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</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="E5:31_1_1"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b3">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">v9</font> = <font color="Maroon" title="c5">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</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="E6:31_1_1"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 for <font color="Olive" title="b1">a</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">w</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">w</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <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">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">w</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">v9</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c7">w9</font> = <font color="Maroon" title="c5">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w9</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">w9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E29">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">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">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)<br/>  for <font color="Olive" title="b3">v9</font>, <font color="Olive" title="b4">w9</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v9</font> &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w9</font> holds <br/>( <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b4">w9</font> &amp; <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v9</font> &amp; <font color="Olive" title="b1">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b4">w9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <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">v9</font>, <font color="Olive" title="b2">w9</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v9</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w9</font> holds <br/>( <font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">w9</font> &amp; <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v9</font> &amp; <font color="Maroon" title="c4">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b2">w9</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">v9</font>, <font color="Maroon" title="c7">w9</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> &amp; <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w9</font> implies ( <font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w9</font> &amp; <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font> &amp; <font color="Maroon" title="c4">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c7">w9</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_1_1_7"/><span class="lab"><font color="Green" title="E30">A9</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font>
 <span class="kw">and </span><br/><a NAME="E2:31_1_1_7"/><span class="lab"><font color="Green" title="E31">A10</font></span>: 
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w9</font>
 ; <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="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w9</font> &amp; <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font> &amp; <font color="Maroon" title="c4">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c7">w9</font> )</span><br/><span class="kw">thus </span><a NAME="E3:31_1_1_7"/>
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">w9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1_7"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E2:31_1_1_7"><span class="lab"><font color="Green" title="E31">A10</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"> ( <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font> &amp; <font color="Maroon" title="c4">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c7">w9</font> )</span><br/><span class="kw">thus </span><a NAME="E4:31_1_1_7"/>
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1_7"><span class="lab"><font color="Green" title="E30">A9</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"> <font color="Maroon" title="c4">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c7">w9</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c4">v</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">w</font> = 
 the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c5">w</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="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c6">v9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c7">w9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1_7"><span class="lab"><font color="Green" title="E30">A9</font></span></a>, <a class="txt" href="#E2:31_1_1_7"><span class="lab"><font color="Green" title="E31">A10</font></span></a>, <a class="ref" href="bhsp_1.html#D1" title="BHSP_1:def.1">BHSP_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><span class="kw">end;</span></div>
<a NAME="E8:31_1_1"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
 for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">w</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <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">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">v9</font> = <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c6">w9</font> = <font color="Maroon" title="c4">w</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">v</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">w</font> = 
<font color="Maroon" title="c6">w9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c5">v9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">w</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c3">v</font>

; <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="E9:31_1_1"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</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">V</font>
 
;<br/>
<a NAME="E10:31_1_1"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #)<br/>  for <font color="Olive" title="b4">a</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) ) &amp; ( <font color="Olive" title="b1">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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) implies <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b4">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Olive" title="b2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <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/> ( ( <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) ) &amp; ( <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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) implies <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="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="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> &amp; <span class="p1">(<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> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">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"> ( ( <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) ) &amp; ( <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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) implies <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="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="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> &amp; <span class="p1">(<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> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font></span>)</span> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c7">x9</font> = <font color="Maroon" title="c3">x</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y9</font> = <font color="Maroon" title="c4">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">z9</font> = <font color="Maroon" title="c5">z</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="lab"><font color="Green" title="E33">A14</font></span>: <span class="p1">(<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> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x9</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y9</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c9">z9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">x9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c9">z9</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c9">z9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></span>
;<br/>
<a NAME="E5:31_1_1_9"/>
<font color="Maroon" title="c7">x9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c7">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:31_1_1_9"/>
( <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <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> <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31_1_1"><span class="lab"><font color="Green" title="E31">A12</font></span></a>, <a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></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="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> &amp; <span class="p1">(<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> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font></span>)</span> )</span><br/>

<a NAME="E7:31_1_1_9"/>
<font color="Maroon" title="c7">x9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c7">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31_1_1_9"/>
 <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="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</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="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font> &amp; <span class="p1">(<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> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font></span>)</span> )</span><br/>

<a NAME="E9:31_1_1_9"/>
<font color="Maroon" title="c7">x9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c8">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:31_1_1_9"/>
<font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</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"> ( <!--HUMANRECFAILED--><span class="p1">(<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> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font></span>)</span> )</span><br/>

<a NAME="E11:31_1_1_9"/>
<font color="Maroon" title="c7">x9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c9">z9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:31_1_1_9"/>
<span class="p1">(<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> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c5">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a>, <a class="txt" href="#E4:31_1_1_9"><span class="lab"><font color="Green" title="E33">A14</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"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font></span>)</span></span><br/>

<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x9</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c8">y9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x9</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Maroon" title="c8">y9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:31_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="bhsp_1.html#K1" title="BHSP_1:func.1">.|.</a> <font color="Maroon" title="c4">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A8</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>
<a NAME="E11:31_1_1"/>
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <br/>  for <font color="Olive" title="b3">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b3">v</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c6">v9</font> = <font color="Maroon" title="c5">v</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font>

<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">W</font> =  <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) as   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31_1_1"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E6:31_1_1"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="txt" href="#E5:31_1_1"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E4:31_1_1"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E10:31_1_1"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="ref" href="bhsp_1.html#D2" title="BHSP_1:def.2">BHSP_1:def 2</a>, <a class="ref" href="rlvect_1.html#D2" title="RLVECT_1:def.2">RLVECT_1:def 2</a>, <a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a>, <a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a>, <a class="ref" href="rlvect_1.html#D5" title="RLVECT_1:def.5">RLVECT_1:def 5</a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a>, <a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>;<br/>
<a NAME="E13:31_1_1"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
(  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font> &amp;  the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">W</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="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T19" title="RELSET_1:th.19">RELSET_1:19</a></span>;<br/>
<a NAME="E14:31_1_1"/>
(  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp;  the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c3">W</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="c1">V</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T19" title="RELSET_1:th.19">RELSET_1:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:31_1_1"/>
 <a href="bhsp_1.html#G1" title="BHSP_1:aggr.1">UNITSTR</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c1">V</font>, the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font>, the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font>, the <a href="bhsp_1.html#U1" title="BHSP_1:sel.1">scalar</a> of <font color="Maroon" title="c1">V</font> #) is   <a href="bhsp_1.html#V1" title="BHSP_1:attr.1">strict</a>   <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:31_1_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="ref" href="rusub_1.html#D1" target="_self" title="RUSUB_1:def.1">Def1</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>
