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<a NAME="E1:30_1_1"/><i><font color="Green" title="E19">A2</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> )
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/>
<a NAME="E2:30_1_1"/><i><font color="Green" title="E20">A3</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</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="c2">U1</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</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">U2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="txt" href="unialg_2.html#E1:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="unialg_2.html#E1:30_1_1"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<div><i><font color="Green" title="E21">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">o</font> be   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</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">C</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c5">o</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">s</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c4">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c5">o</font> implies <font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )</span><br/><b>assume </b><a NAME="E1:30_1_1_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c5">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font></span><br/><b>set </b><font color="Maroon" title="c7">B1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font>;<br/><b>set </b><font color="Maroon" title="c8">B2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">s1</font> = <font color="Maroon" title="c6">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> <b>by </b><i><a class="txt" href="unialg_2.html#E2:30_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">B1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon" title="c10">B1</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 <b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/><a NAME="E5:30_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">B1</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c5">o</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></i>;<br/><a NAME="E6:30_1_1_1_1"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">B1</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:30_1_1_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">s2</font> = <font color="Maroon" title="c6">s</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font> <b>by </b><i><a class="txt" href="unialg_2.html#E2:30_1_1"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="finseq_2.html#T27" title="FINSEQ_2:th.27">FINSEQ_2:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">B2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/><a NAME="E9:30_1_1_1_1"/>
<font color="Maroon" title="c12">B2</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 <b>by </b><i><a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/><a NAME="E10:30_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">B2</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c5">o</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></i>;<br/><a NAME="E11:30_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">B2</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E1:30_1_1_1_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E12:30_1_1_1_1"/>
<font color="Maroon" title="c5">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E6:30_1_1_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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>hence </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon" title="c4">C</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c5">o</font>
 <b>by </b><i><a class="ref" href="unialg_2.html#D4" target="_self" title="UNIALG_2: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>set </b><font color="Maroon" title="c5">S</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c4">C</font></span>)</span> #);<br/>
<a NAME="E5:30_1_1"/><i><font color="Green" title="E22">A7</font></i>: 
for <font color="Olive" title="b1">B</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Olive" title="b1">B</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 <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c4">C</font></span>)</span> #) holds <br/>( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c4">C</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E4:30_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">S</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c4">C</font>,<span class="p1">(<span class="default"><a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Maroon" title="c4">C</font></span>)</span> #) as   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a> <b>by </b><i><a class="ref" href="unialg_2.html#T17" target="_self" title="UNIALG_2:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c6">S</font> as  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>  <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font> <b>by </b><i><a class="txt" href="unialg_2.html#E5:30_1_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="unialg_2.html#D8" target="_self" title="UNIALG_2:def.8">Def8</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">S</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="c7">S</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="c2">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font> &amp; ( for <font color="Olive" title="b1">B</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Olive" title="b1">B</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="c7">S</font> holds <br/>( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> ) ) )</span><br/>

<b>thus </b><a NAME="E8:30_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S</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="c2">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">U2</font> &amp; ( for <font color="Olive" title="b1">B</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">U0</font>  st <font color="Olive" title="b1">B</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="c7">S</font> holds <br/>( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c7">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K3" title="UNIALG_2:func.3">Opers</a> <font color="Maroon" title="c1">U0</font>,<font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a> ) ) )
 <b>by </b><i><a class="txt" href="unialg_2.html#E4:30_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</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>
