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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font>,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A2</font> holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Olive" title="b2">A2</font>;<br/>
<a NAME="E1:26_1_1"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 for <font color="Olive" title="b1">W1</font>, <font color="Olive" title="b2">W2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b3">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">W1</font>,<font color="Olive" title="b2">W2</font>,<font color="Olive" title="b3">V</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</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">  ex <font color="Olive" title="b1">V</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Olive" title="b1">V</font>]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">A1</font> = <font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c6">A2</font> = <font color="Maroon" title="c4">W2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T15" target="_self" onmouseover="rs('group_6/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c5">A1</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c6">A2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T16" target="_self" onmouseover="rs('group_6/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">C</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">S<sub>1</sub></font>[<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c7">C</font>]</span><br/>

<span class="kw">thus </span><a NAME="E3:26_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">W1</font>,<font color="Maroon" title="c4">W2</font>,<font color="Maroon" title="c7">C</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">thus </span><a NAME="E2:26_1_1"/>
 ex <font color="Olive" title="b1">B</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font></span>)</span> st <br/> for <font color="Olive" title="b2">W1</font>, <font color="Olive" title="b3">W2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_6.html#NK4" title="GROUP_6:NK.4">Cosets</a> <font color="Maroon" title="c2">N</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</font>,<font color="Olive" title="b1">B</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Olive" title="b2">W1</font>,<font color="Olive" title="b3">W2</font>)]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="binop_1.html#S3" onmouseover="rs('binop_1/S3')" onmouseout="rh()">BINOP_1:sch 3</a>(<a class="txt" href="#E1:26_1_1"><span class="lab"><font color="Green" title="E17">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/>


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