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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be    <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</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> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( <font color="Olive" title="b1">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Olive" title="b1">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</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="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  iff  <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span><br/>



<div><span class="kw">hereby </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"> (  <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  implies <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:46_1"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> </span><br/><a NAME="E2:46_1"/><span class="kw">then </span>
 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="c1">M</font>  st <font color="Olive" title="b1">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/><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="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">A9</font> = <font color="Maroon" title="c2">A</font> as   <a href="algstr_4.html#V2" title="ALGSTR_4:attr.2">stable</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D10" target="_self" onmouseover="rs('algstr_4/D10')" onmouseout="rh()">Def10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">N</font> =  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# <font color="Maroon" title="c3">A9</font>,<span class="p1">(<span class="default"><a href="algstr_4.html#K8" title="ALGSTR_4:func.8">the_mult_induced_by</a> <font color="Maroon" title="c3">A9</font></span>)</span> #) as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</a></span>;<br/><a NAME="E5:46_1"/>
 <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font> is    <a href="algstr_4.html#M1" title="ALGSTR_4:mode.1">multSubmagma</a> of <font color="Maroon" title="c4">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D12" target="_self" onmouseover="rs('algstr_4/D12')" onmouseout="rh()">Def12</a></span>;<br/><a NAME="E6:46_1"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D9" target="_self" onmouseover="rs('algstr_4/D9')" onmouseout="rh()">Def9</a></span>;<br/><span class="kw">hence </span><a NAME="E7:46_1"/>
 <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46_1"><span class="lab"><font color="Green" title="E21">A1</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">assume </span><a NAME="E2:46"/>
 <a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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"> <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/>

<a NAME="E3:46"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K9" title="ALGSTR_4:func.9">the_submagma_generated_by</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E4:46"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D12" target="_self" onmouseover="rs('algstr_4/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46"/>
<font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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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