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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="funcsdom.html#NM2" title="FUNCSDOM:NM.2">Algebra</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#V4" title="C0SP1:attr.4">additively-linearly-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="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Olive" title="b1">V1</font>,<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#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#K5" title="C0SP1:func.5">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="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c1">V</font></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#V4" title="C0SP1:attr.4">additively-linearly-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="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c2">V1</font>,<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#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#K5" title="C0SP1:func.5">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="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c1">V</font> )</span><br/>



<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-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="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c2">V1</font>,<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#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#K5" title="C0SP1:func.5">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="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c2">V1</font> is <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> &amp; <font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><br/> for <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> holds <br/><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> ) )
 
<b>by </b><i><a class="ref" href="c0sp1.html#D10" target="_self" title="C0SP1:def.10">def210</a></i>;<br/>
<a NAME="E3:27"/><b>then </b>
(  <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> &amp;  <a href="c0sp1.html#K5" title="C0SP1:func.5">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="rlvect_1.html#U1" title="RLVECT_1:sel.1">Mult</a> of <font color="Maroon" title="c1">V</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c2">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="c0sp1.html#E1:27"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="c0sp1.html#D7" target="_self" title="C0SP1:def.7">Rdef213</a>, <a class="ref" href="c0sp1.html#D5" target="_self" title="C0SP1:def.5">Rdef211</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>, <a class="ref" href="c0sp1.html#D11" target="_self" title="C0SP1:def.11">def212</a></i>;<br/>
<b>hence </b><a NAME="E4:27"/>
 <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c2">V1</font>,<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#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#K5" title="C0SP1:func.5">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="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="txt" href="c0sp1.html#E1:27"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="c0sp1.html#E2:27"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="c0sp1.html#T3" target="_self" title="C0SP1:th.3">Th02</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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