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<b>let </b><font color="Maroon" title="c3">IT</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SF</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  iff <font color="Maroon" title="c3">IT</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">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> <font color="Maroon" title="c1">SF</font> )</span><br/>

<a NAME="E1:22_1_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">x</font> is <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a>
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="algstr_0.html#D39" title="ALGSTR_0:def.39">ALGSTR_0:def 39</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SF</font><b> such that </b><br/><a NAME="E3:22_1_1"/><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c4">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">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> <font color="Maroon" title="c1">SF</font>
 <b>by </b><i><a class="ref" href="algstr_0.html#D27" title="ALGSTR_0:def.27">ALGSTR_0:def 27</a></i>;<br/>
<a NAME="E4:22_1_1"/>
<font color="Maroon" title="c2">x</font> is <a href="algstr_0.html#V24" title="ALGSTR_0:attr.24">right_invertible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c4">y</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D28" title="ALGSTR_0:def.28">ALGSTR_0:def 28</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</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">SF</font></span><br/>

<b>thus </b><a NAME="E1:22_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">y</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">SF</font>
 <b>by </b><i><a class="txt" href="#E3:22_1_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="vectsp_2.html#T41" target="_self" title="VECTSP_2:th.41">Th41</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>then consider </b><font color="Maroon" title="c5">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SF</font><b> such that </b><br/><a NAME="E6:22_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x1</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">SF</font>
 <b>by </b><i><a class="ref" href="algstr_0.html#D28" title="ALGSTR_0:def.28">ALGSTR_0:def 28</a></i>;<br/>
<a NAME="E7:22_1_1"/>
<font color="Maroon" title="c2">x</font> is <a href="algstr_0.html#V17" title="ALGSTR_0:attr.17">right_mult-cancelable</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font>, <font color="Maroon" title="c7">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">SF</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="algstr_0.html#D21" title="ALGSTR_0:def.21">ALGSTR_0:def 21</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> )</span><br/>

<b>assume </b><a NAME="E1:22_1_1_2"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font></span><br/>

<b>thus </b><font color="Maroon" title="c6">y</font> = 
<font color="Maroon" title="c6">y</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="c1">SF</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x1</font>
<b>by </b><i><a class="txt" href="#E6:22_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E1:22_1_1_2"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c7">z</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="c1">SF</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:22_1_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c7">z</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</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="E8:22_1_1"/>
( <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  iff <font color="Maroon" title="c3">IT</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">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> <font color="Maroon" title="c1">SF</font> )
 <b>by </b><i><a class="txt" href="#E1:22_1_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="algstr_0.html#D35" title="ALGSTR_0:def.35">ALGSTR_0:def 35</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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