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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">V1</font> is <a href="c0sp1.html#V2" title="C0SP1:attr.2">additively-closed</a> &amp; <font color="Olive" title="b1">V1</font> is <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a> &amp; not <font color="Olive" title="b1">V1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> holds <br/> <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Olive" title="b1">V1</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K1" title="C0SP1:func.1">Add_</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K4" title="C0SP1:func.4">One_</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K3" title="C0SP1:func.3">Zero_</a> <font color="Olive" title="b1">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a></span><br/><b>let </b><font color="Maroon" title="c2">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V2" title="C0SP1:attr.2">additively-closed</a> &amp; <font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a> &amp; not <font color="Maroon" title="c2">V1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> implies  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K1" title="C0SP1:func.1">Add_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K4" title="C0SP1:func.4">One_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K3" title="C0SP1:func.3">Zero_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> )</span><br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V2" title="C0SP1:attr.2">additively-closed</a> &amp; <font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a> &amp; not <font color="Maroon" title="c2">V1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K1" title="C0SP1:func.1">Add_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K4" title="C0SP1:func.4">One_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K3" title="C0SP1:func.3">Zero_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a></span><br/>

<a NAME="E2:16"/>
(  <a href="c0sp1.html#K3" title="C0SP1:func.3">Zero_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="c0sp1.html#K1" title="C0SP1:func.1">Add_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font> &amp;  <a href="c0sp1.html#K4" title="C0SP1:func.4">One_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">V</font> &amp;  <a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c1">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c2">V1</font> )
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="c0sp1.html#D5" target="_self" title="C0SP1:def.5">Rdef211</a>, <a class="ref" href="c0sp1.html#D7" target="_self" title="C0SP1:def.7">Rdef213</a>, <a class="ref" href="c0sp1.html#D8" target="_self" title="C0SP1:def.8">Rdef214</a>, <a class="ref" href="c0sp1.html#D6" target="_self" title="C0SP1:def.6">Rdef220</a></i>;<br/>
<b>hence </b><a NAME="E3:16"/>
 <a href="algstr_0.html#G6" title="ALGSTR_0:aggr.6">doubleLoopStr</a>(# <font color="Maroon" title="c2">V1</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K1" title="C0SP1:func.1">Add_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K4" title="C0SP1:func.4">One_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span>,<span class="p1">(<span class="default"><a href="c0sp1.html#K3" title="C0SP1:func.3">Zero_</a> <font color="Maroon" title="c2">V1</font>,<font color="Maroon" title="c1">V</font></span>)</span> #) is   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="c0sp1.html#T1" target="_self" title="C0SP1:th.1">Th01</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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