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<div class="add">

<b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</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">B</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="polyalg1.html#L1" title="POLYALG1:struct.1">AlgebraStr</a> of <font color="Maroon" title="c2">L</font><br/> for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">B</font>  st <font color="Olive" title="b2">A</font> is <a href="polyalg1.html#V3" title="POLYALG1:attr.3">opers_closed</a> holds <br/> ex <font color="Olive" title="b3">C</font> being <a href="polyalg1.html#V1" title="POLYALG1:attr.1">strict</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Olive" title="b1">B</font> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">A</font></span><br/><b>let </b><font color="Maroon" title="c3">B</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <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"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">B</font>  st <font color="Olive" title="b1">A</font> is <a href="polyalg1.html#V3" title="POLYALG1:attr.3">opers_closed</a> holds <br/> ex <font color="Olive" title="b2">C</font> being <a href="polyalg1.html#V1" title="POLYALG1:attr.1">strict</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font></span><br/><b>let </b><font color="Maroon" title="c4">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is <a href="polyalg1.html#V3" title="POLYALG1:attr.3">opers_closed</a> implies  ex <font color="Olive" title="b1">C</font> being <a href="polyalg1.html#V1" title="POLYALG1:attr.1">strict</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> )</span><br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="polyalg1.html#V3" title="POLYALG1:attr.3">opers_closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">C</font> being <a href="polyalg1.html#V1" title="POLYALG1:attr.1">strict</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font></span><br/>

<a NAME="E2:43"/><i><font color="Green" title="E23">A2</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c3">B</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f1</font> = the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c3">B</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">A</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E4:43"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f1</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"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<a NAME="E1:43_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E2:43"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="polyalg1.html#T1" target="_self" title="POLYALG1:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E2:43_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f1</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"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>; <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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/>

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<b>assume </b><a NAME="E3:43_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:43_1"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E3:43_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c10">z</font> = <font color="Maroon" title="c8">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="#E5:43_1"><i><font color="Green" title="E25">A4</font></i></a></i>;<br/>
<a NAME="E7:43_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a>
 
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="polyalg1.html#D4" target="_self" title="POLYALG1:def.4">Def4</a></i>;<br/>
<a NAME="E8:43_1"/>
<font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c10">z</font>
 
<b>by </b><i><a class="txt" href="#E3:43_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="#E5:43_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1"/>
<font color="Maroon" title="c5">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="#E5:43_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="#E7:43_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">VECTSP_4:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">ad</font> = <font color="Maroon" title="c5">f1</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c4">A</font> <b>by </b><i><a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">f2</font> = the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c3">B</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">A</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E7:43"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E2:43"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="polyalg1.html#T1" target="_self" title="POLYALG1:th.1">Th1</a></i>;<br/>
<a NAME="E8:43"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">f2</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"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E9:43"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/>

<b>consider </b><font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_2"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:43_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c12">z</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="#E3:43_2"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>
<a NAME="E5:43_2"/>
<font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">z</font>
 
<b>by </b><i><a class="txt" href="#E1:43_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="#E3:43_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E6:43_2"/>
<font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="#E3:43_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="polyalg1.html#D4" target="_self" title="POLYALG1:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">mu</font> = <font color="Maroon" title="c7">f2</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c4">A</font> <b>by </b><i><a class="txt" href="#E8:43"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">f4</font> = the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <font color="Maroon" title="c3">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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E12:43"/>
<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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E13:43"/><b>then </b>
<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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c3">B</font>
 
<b>by </b><i><a class="ref" href="polyalg1.html#T2" target="_self" title="POLYALG1:th.2">Th2</a></i>;<br/>
<a NAME="E14:43"/><b>then </b><i><font color="Green" title="E25">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">f4</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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E15:43"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c9">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <font color="Maroon" title="c9">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )</span><br/>

<b>assume </b><a NAME="E1:43_3"/><i><font color="Green" title="E26">A10</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font></span><br/>

<b>consider </b><font color="Maroon" title="c11">y</font>, <font color="Maroon" title="c12">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:43_3"/><i><font color="Green" title="E27">A11</font></i>: 
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">y</font>,<font color="Maroon" title="c12">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:43_3"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> <b>by </b><i><a class="txt" href="#E3:43_3"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c12">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="#E3:43_3"><i><font color="Green" title="E27">A11</font></i></a></i>;<br/>
<a NAME="E6:43_3"/><i><font color="Green" title="E28">A12</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a>
 
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="polyalg1.html#D4" target="_self" title="POLYALG1:def.4">Def4</a></i>;<br/>
<a NAME="E7:43_3"/>
<font color="Maroon" title="c9">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c14">z</font>
 
<b>by </b><i><a class="txt" href="#E1:43_3"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="#E3:43_3"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E8:43_3"/>
<font color="Maroon" title="c9">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font>
 <b>by </b><i><a class="txt" href="#E3:43_3"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="#E6:43_3"><i><font color="Green" title="E28">A12</font></i></a>, <a class="ref" href="vectsp_4.html#D1" title="VECTSP_4:def.1">VECTSP_4:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">lm</font> = <font color="Maroon" title="c9">f4</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>,<font color="Maroon" title="c4">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">A</font> <b>by </b><i><a class="txt" href="#E14:43"><i><font color="Green" title="E25">A9</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">z</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c3">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="polyalg1.html#D4" target="_self" title="POLYALG1:def.4">Def4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">u</font> =  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c3">B</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">A</font> <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="polyalg1.html#D4" target="_self" title="POLYALG1:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">c</font> =  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">ad</font>,<font color="Maroon" title="c8">mu</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c10">lm</font> #);<br/>
<a NAME="E19:43"/><i><font color="Green" title="E26">A13</font></i>: 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">ad</font>,<font color="Maroon" title="c8">mu</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c10">lm</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="c3">B</font>
 
;<br/>
<a NAME="E20:43"/>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">ad</font>,<font color="Maroon" title="c8">mu</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c10">lm</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="c3">B</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">C</font> =  <a href="polyalg1.html#G1" title="POLYALG1:aggr.1">AlgebraStr</a>(# <font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c6">ad</font>,<font color="Maroon" title="c8">mu</font>,<font color="Maroon" title="c11">z</font>,<font color="Maroon" title="c12">u</font>,<font color="Maroon" title="c10">lm</font> #) as  <a href="polyalg1.html#V1" title="POLYALG1:attr.1">strict</a>  <a href="polyalg1.html#M1" title="POLYALG1:mode.1">Subalgebra</a> of <font color="Maroon" title="c3">B</font> <b>by </b><i><a class="txt" href="#E19:43"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="polyalg1.html#D3" target="_self" title="POLYALG1:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font></span><br/>

<b>thus </b><a NAME="E22:43"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c14">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
