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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  implies ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_5_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">G</font> = <font color="Maroon" title="c1">G</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a>   <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ;<br/>
<a NAME="E3:37_5_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>  st ( <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> <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> <font color="Olive" title="b3">c</font> or <font color="Olive" title="b2">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">a</font> ) holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T14" title="GROUP_1:th.14">GROUP_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:37_5_1"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  &amp; <font color="Maroon" title="c1">G</font> is  <a href="monoid_0.html#V16" title="MONOID_0:attr.16">cancelable</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#T15" target="_self" title="MONOID_0:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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