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<span class="kw">let </span><font color="Maroon" title="c2">V</font> be   <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Algebra</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">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">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="c2">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="c2">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="c2">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="c2">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="c2">V</font>)</span>)</span> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">V</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="c3">V1</font> is  <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a>  &amp; <font color="Maroon" title="c3">V1</font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a>  &amp;  not <font color="Maroon" title="c3">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="c3">V1</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">V</font>)</span>)</span> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c2">V</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
( <font color="Maroon" title="c3">V1</font> is  <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a>  &amp; <font color="Maroon" title="c3">V1</font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a>  &amp;  not <font color="Maroon" title="c3">V1</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="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c3">V1</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">V</font>)</span>)</span> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c2">V</font></span><br/>

<a NAME="E2:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 <a href="c0sp1.html#K5" title="C0SP1:func.5">Mult_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c2">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="c3">V1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D11" target="_self" onmouseover="rs('c0sp1/D11')" onmouseout="rh()">Def11</a></span>;<br/>
<a NAME="E3:25"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
(  <a href="c0sp1.html#K4" title="C0SP1:func.4">One_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c2">V</font> &amp;  <a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="c0sp1.html#D6" target="_self" onmouseover="rs('c0sp1/D6')" onmouseout="rh()">Def6</a>, <a class="ref" href="c0sp1.html#D8" target="_self" onmouseover="rs('c0sp1/D8')" onmouseout="rh()">Def8</a></span>;<br/>
<a NAME="E4:25"/>
(  <a href="c0sp1.html#K3" title="C0SP1:func.3">Zero_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c2">V</font> &amp;  <a href="c0sp1.html#K1" title="C0SP1:func.1">Add_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c2">V</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">V1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="c0sp1.html#D5" target="_self" onmouseover="rs('c0sp1/D5')" onmouseout="rh()">Def5</a>, <a class="ref" href="c0sp1.html#D7" target="_self" onmouseover="rs('c0sp1/D7')" onmouseout="rh()">Def7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:25"/>
 <a href="funcsdom.html#G1" title="FUNCSDOM:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c3">V1</font>,<span class="p1">(<span class="default"><a href="c0sp1.html#K2" title="C0SP1:func.2">mult_</a> (<font color="Maroon" title="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">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="c3">V1</font>,<font color="Maroon" title="c2">V</font>)</span>)</span> #) is    <a href="c0sp1.html#M2" title="C0SP1:mode.2">Subalgebra</a> of <font color="Maroon" title="c2">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="c0sp1.html#T3" target="_self" onmouseover="rs('c0sp1/T3')" onmouseout="rh()">Th3</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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