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<div class="add">

<b>let </b><font color="Maroon" title="c1">W</font> be    <a href="lopban_2.html#L1" title="LOPBAN_2:struct.1">Normed_AlgebraStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a>  st  <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">W</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">W</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">W</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">W</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">U3</a> of <font color="Maroon" title="c1">W</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">W</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">V</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">W</font> holds <br/><font color="Maroon" title="c1">W</font> is   <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a></span><br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">W</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">W</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">W</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">W</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">U3</a> of <font color="Maroon" title="c1">W</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">W</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">W</font> implies <font color="Maroon" title="c1">W</font> is   <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a> )</span><br/>



<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E36">A1</font></i>: 
(  <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">W</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">W</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">W</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">W</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">U3</a> of <font color="Maroon" title="c1">W</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">U2</a> of <font color="Maroon" title="c1">W</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">V</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">W</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">W</font> is   <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c3">W</font> = <font color="Maroon" title="c1">W</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="funcsdom.html#L1" title="FUNCSDOM:struct.1">AlgebraStr</a>  <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:53"/><i><font color="Green" title="E37">L1</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">W</font> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y1</font> = <font color="Maroon" title="c5">y</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c7">y1</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:53_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">x1</font>
 
;<br/>
<b>hence </b><a NAME="E4:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:53"/><i><font color="Green" title="E38">L2</font></i>: 
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 <font color="Maroon" title="c3">W</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">z</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">z1</font> = <font color="Maroon" title="c6">z</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">z1</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:53_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">z1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:53_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:53"/><i><font color="Green" title="E39">L3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">W</font> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c6">z</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">W</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_3"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</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="c2">V</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:53_3"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:53"/><i><font color="Green" title="E40">L4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font> holds  <font color="Olive" title="b1">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font><b> such that </b><br/><a NAME="E3:53_4"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c5">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">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="c2">V</font>
 <b>by </b><i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">v</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E5:53_4"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">W</font>
 
<b>by </b><i><a class="txt" href="#E3:53_4"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:53_4"/>
<font color="Maroon" title="c4">x</font> is <a href="algstr_0.html#V10" title="ALGSTR_0:attr.10">right_complementable</a>
 <b>by </b><i><a class="ref" href="algstr_0.html#D11" title="ALGSTR_0:def.11">ALGSTR_0:def 11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E7:53"/><i><font color="Green" title="E41">L5</font></i>: 
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font> holds  <font color="Olive" title="b1">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c5">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c6">v1</font> = <font color="Maroon" title="c4">v</font>, <font color="Maroon" title="c7">w1</font> = <font color="Maroon" title="c5">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_5"/>
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">w1</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:53_5"/><b>then </b>
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c6">v1</font>
 
;<br/>
<b>hence </b><a NAME="E4:53_5"/>
<font color="Maroon" title="c4">v</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">w</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">v</font>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E8:53"/><i><font color="Green" title="E42">L6</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">x</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c8">a1</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c9">b1</font> = <font color="Maroon" title="c5">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">b1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:53_6"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">b1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c7">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:53_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E9:53"/><i><font color="Green" title="E43">L7</font></i>: 
<font color="Maroon" title="c3">W</font> is <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_7"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">V</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:53_7"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">W</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E10:53"/><i><font color="Green" title="E44">L8</font></i>: 
<font color="Maroon" title="c3">W</font> is <a href="vectsp_1.html#V1" title="VECTSP_1:attr.1">right-distributive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">z1</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_8"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">z1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:53_8"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">y1</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="c7">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c9">z1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp_1.html#D11" title="VECTSP_1:def.11">VECTSP_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E4:53_8"/>
<font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</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">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E11:53"/>
<font color="Maroon" title="c3">W</font> is <a href="funcsdom.html#V2" title="FUNCSDOM:attr.2">Algebra-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">W</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  holds <br/> ( <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font> &amp; <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c11">z1</font> = <font color="Maroon" title="c6">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">V</font> <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E2:53_9"/>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">y1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E3:53_9"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c10">y1</font>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E4:53_9"/>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">y</font>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/>

<a NAME="E5:53_9"/>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E6:53_9"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x1</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="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E7:53_9"/>
<font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/>

<a NAME="E8:53_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x1</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E9:53_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x1</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="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E10:53_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/>

<a NAME="E11:53_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x1</font>
 
<b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E12:53_9"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funcsdom.html#D20" title="FUNCSDOM:def.20">FUNCSDOM:def 20</a></i>;<br/>
<b>hence </b><a NAME="E13:53_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:53"><i><font color="Green" title="E36">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:53"/>
<font color="Maroon" title="c1">W</font> is   <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</a>
 <b>by </b><i><a class="txt" href="#E3:53"><i><font color="Green" title="E37">L1</font></i></a>, <a class="txt" href="#E4:53"><i><font color="Green" title="E38">L2</font></i></a>, <a class="txt" href="#E5:53"><i><font color="Green" title="E39">L3</font></i></a>, <a class="txt" href="#E6:53"><i><font color="Green" title="E40">L4</font></i></a>, <a class="txt" href="#E7:53"><i><font color="Green" title="E41">L5</font></i></a>, <a class="txt" href="#E8:53"><i><font color="Green" title="E42">L6</font></i></a>, <a class="txt" href="#E9:53"><i><font color="Green" title="E43">L7</font></i></a>, <a class="txt" href="#E10:53"><i><font color="Green" title="E44">L8</font></i></a>, <a class="ref" href="algstr_0.html#D16" title="ALGSTR_0:def.16">ALGSTR_0:def 16</a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a>, <a class="ref" href="group_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
