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<b>let </b><font color="Maroon" title="c2">L</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</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">A</font>, <font color="Olive" title="b2">B</font> being   <a href="polyalg1.html#L1" title="POLYALG1:struct.1">AlgebraStr</a> of <font color="Maroon" title="c2">L</font>  st <font color="Olive" title="b1">A</font> is    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b2">B</font> is    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Olive" title="b1">A</font> holds <br/> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">A</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">A</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">A</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Olive" title="b1">A</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Olive" title="b1">A</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Olive" title="b1">A</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">B</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b2">B</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b2">B</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Olive" title="b2">B</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Olive" title="b2">B</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Olive" title="b2">B</font> #)</span><br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="polyalg1.html#L1" title="POLYALG1:struct.1">AlgebraStr</a> of <font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> is    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c4">B</font> &amp; <font color="Maroon" title="c4">B</font> is    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">A</font> implies  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">A</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">A</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c3">A</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c3">A</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">A</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">B</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c4">B</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">B</font> #) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">A</font> is    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c4">B</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c4">B</font> is    <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">A</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">A</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c3">A</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c3">A</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">A</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">B</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c4">B</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">B</font> #)</span><br/>

<a NAME="E3:33"/><i><font color="Green" title="E18">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> <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">B</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green" title="E19">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font> <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="c3">A</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E2:33"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/>
<a NAME="E5:33"/><b>then </b><i><font color="Green" title="E20">A5</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E3:33"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:33"/><i><font color="Green" title="E21">A6</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="polyalg1.html#T1" target="_self" title="POLYALG1:th.1">Th1</a></i>;<br/>
<a NAME="E7:33"/><i><font color="Green" title="E22">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="polyalg1.html#T1" target="_self" title="POLYALG1:th.1">Th1</a></i>;<br/>
<a NAME="E8:33"/><i><font color="Green" title="E23">A8</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="polyalg1.html#T2" target="_self" title="POLYALG1:th.2">Th2</a></i>;<br/>
<i><font color="Green" title="E24">A9</font></i>: the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">A</font> = 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">B</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="txt" href="polyalg1.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>
<br/>.= 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="txt" href="polyalg1.html#E5:33"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="polyalg1.html#E6:33"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>
;<br/>
<i><font color="Green" title="E25">A10</font></i>: the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">A</font> = 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">B</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>
<b>by </b><i><a class="txt" href="polyalg1.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>
<br/>.= 
the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="txt" href="polyalg1.html#E5:33"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="polyalg1.html#E7:33"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>
;<br/>
<i><font color="Green" title="E26">A11</font></i>: the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">A</font> = 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">B</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">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="polyalg1.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>
<br/>.= 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">B</font>
<b>by </b><i><a class="txt" href="polyalg1.html#E4:33"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="polyalg1.html#E8:33"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>
;<br/>
<a NAME="E12:33"/><i><font color="Green" title="E27">A12</font></i>: 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">A</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="c4">B</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/>
<a NAME="E13:33"/>
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c4">B</font>
 
<b>by </b><i><a class="txt" href="polyalg1.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/>
<b>hence </b><a NAME="E14:33"/>
 <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">A</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">A</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">A</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c3">A</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c3">A</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">A</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">B</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c4">B</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">ZeroF</a> of <font color="Maroon" title="c4">B</font>,the <a href="struct_0.html#U3" title="STRUCT_0:sel.3">OneF</a> of <font color="Maroon" title="c4">B</font>,the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c4">B</font> #)
 <b>by </b><i><a class="txt" href="polyalg1.html#E3:33"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="polyalg1.html#E4:33"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="polyalg1.html#E9:33"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="polyalg1.html#E10:33"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="polyalg1.html#E11:33"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="polyalg1.html#E12:33"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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