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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="vectsp_1.html#V3" title="VECTSP_1:attr.3">right_unital</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">b</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a>  &amp; <font color="Olive" title="b2">b</font> is  <a href="algstr_0.html#V17" title="ALGSTR_0:attr.17">right_mult-cancelable</a>  &amp; <font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</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="b2">b</font> holds <br/><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#K12" title="ALGSTR_0:func.12">/</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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="c3">b</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a>  &amp; <font color="Maroon" title="c3">b</font> is  <a href="algstr_0.html#V17" title="ALGSTR_0:attr.17">right_mult-cancelable</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
( <font color="Maroon" title="c3">b</font> is  <a href="algstr_0.html#V23" title="ALGSTR_0:attr.23">left_invertible</a>  &amp; <font color="Maroon" title="c3">b</font> is  <a href="algstr_0.html#V17" title="ALGSTR_0:attr.17">right_mult-cancelable</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"> (  not <font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:10"/>
<font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</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"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E3:10"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</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="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="ref" href="algstr_0.html#D30" onmouseover="rs('algstr_0/D30')" onmouseout="rh()">ALGSTR_0:def 30</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c2">a</font> = 
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K9" title="ALGSTR_0:func.9">/</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c3">b</font>
<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/> <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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