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

<span class="kw">set </span><font color="Maroon" title="c3">P</font> =  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font>;<br/>
<a NAME="E1:100_1_1"/><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> <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="c2">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D14" target="_self" onmouseover="rs('field_7/D14')" onmouseout="rh()">d</a></span>;<br/>
<a NAME="E2:100_1_1"/><span class="lab"><font color="Green" title="E52">A2</font></span>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> <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">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D14" target="_self" onmouseover="rs('field_7/D14')" onmouseout="rh()">d</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> holds  <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E63"><span class="lab"><font color="Green" title="E48">Lm10</font></span></a></span>;<br/><span class="kw">thus </span><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> = 
<font color="Maroon" title="c6">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D12" onmouseover="rs('group_1/D12')" onmouseout="rh()">GROUP_1:def 12</a></span>
<br/>.= 
<font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:100_1_1"/>
 <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp;  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>  &amp;  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  &amp;  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span></span></a><div class="add"><span class="kw">let </span><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 <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="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#K6" title="ALGSTR_0:func.6">*</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#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#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">c</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">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E63"><span class="lab"><font color="Green" title="E48">Lm10</font></span></a></span>;<br/><a NAME="E2:100_1_1_2"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>;<br/><a NAME="E3:100_1_1_2"/>
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>;<br/><span class="kw">hence </span><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#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<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#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100_1_1_2"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:100_1_1"/>
 <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>  &amp;  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  &amp;  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">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> <span class="p2">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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> <span class="p2">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">a</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">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E63"><span class="lab"><font color="Green" title="E48">Lm10</font></span></a></span>;<br/><span class="kw">thus </span><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> <span class="p2">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c5">a</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="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p2">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">x</font>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:100_1_1"/>
 <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  &amp;  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">z</font></span>)</span> <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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font></span>)</span> )</span></a><div class="add"><span class="kw">let </span><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 <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</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="c6">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font></span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">c</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">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E63"><span class="lab"><font color="Green" title="E48">Lm10</font></span></a></span>;<br/><a NAME="E2:100_1_1_4"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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 class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>;<br/><a NAME="E3:100_1_1_4"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
( <font color="Maroon" title="c8">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c9">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>;<br/><a NAME="E4:100_1_1_4"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E64"><span class="lab"><font color="Green" title="E49">Lm11</font></span></a></span>;<br/><span class="kw">hence </span><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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> = 
<font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D7" onmouseover="rs('vectsp_1/D7')" onmouseout="rh()">VECTSP_1:def 7</a></span>
<br/>.= 
<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="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:100_1_1_4"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="field_7.html#E64"><span class="lab"><font color="Green" title="E49">Lm11</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</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="c6">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">c</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100_1_1_4"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">a</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="c9">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D7" onmouseover="rs('vectsp_1/D7')" onmouseout="rh()">VECTSP_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</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="c6">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:100_1_1_4"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="field_7.html#E64"><span class="lab"><font color="Green" title="E49">Lm11</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:100_1_1"/>
 <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1_1_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> st <font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> implies  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> st <font color="Olive" title="b1">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:100_1_1_5"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> st <font color="Olive" title="b1">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c5">a</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">E</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E63"><span class="lab"><font color="Green" title="E48">Lm10</font></span></a></span>;<br/><a NAME="E3:100_1_1_5"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="field_7.html#K11" title="FIELD_7:func.11">Alg_El</a> <font color="Maroon" title="c2">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D14" target="_self" onmouseover="rs('field_7/D14')" onmouseout="rh()">d</a></span>;<br/><a NAME="E4:100_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="field_7.html#K11" title="FIELD_7:func.11">Alg_El</a> <font color="Maroon" title="c2">E</font>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x1</font> being  <font color="Maroon" title="c1">F</font> <a href="field_6.html#V2" title="FIELD_6:attr.2">-algebraic</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">E</font><span class="kw"> such that </span><br/><a NAME="E6:100_1_1_5"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font>
 ;<br/><a NAME="E7:100_1_1_5"/>
 not <font color="Maroon" title="c6">x1</font> is  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D14" target="_self" onmouseover="rs('field_7/D14')" onmouseout="rh()">d</a>, <a class="txt" href="#E1:100_1_1_5"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E6:100_1_1_5"><span class="lab"><font color="Green" title="E54">A6</font></span></a></span>;<br/><a NAME="E8:100_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="field_7.html#K11" title="FIELD_7:func.11">Alg_El</a> <font color="Maroon" title="c2">E</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_7.html#D14" target="_self" onmouseover="rs('field_7/D14')" onmouseout="rh()">d</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span></span><br/><span class="kw">thus </span><font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="field_7.html#E65"><span class="lab"><font color="Green" title="E50">Lm12</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E2:100_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E1:100_1_1_5"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E6:100_1_1_5"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="vectsp_1.html#D10" onmouseover="rs('vectsp_1/D10')" onmouseout="rh()">VECTSP_1:def 10</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:100_1_1"/>
 <a href="field_7.html#NK13" title="FIELD_7:NK.13">F_Alg</a> <font color="Maroon" title="c2">E</font> is  <a href="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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