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<b>let </b><font color="Maroon" title="c1">U0</font> be   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a>;<br/><b>let </b><font color="Maroon" title="c2">U1</font> be    <a href="unialg_2.html#M3" title="UNIALG_2:mode.3">SubAlgebra</a> of <font color="Maroon" title="c1">U0</font>;<br/>



<b>set </b><font color="Maroon" title="c3">u1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">U1</font>;<br/>
<a NAME="E1:35"/>
 <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">U0</font><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp;  ex <font color="Olive" title="b1">o</font> being  <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font> st <br/>(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">o</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">o0</font> being   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c1">U0</font><b> such that </b><br/><a NAME="E5:35_1"/><i><font color="Green" title="E22">A2</font></i>: 
(  <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">o0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">o0</font> )
 <b>by </b><i><a class="txt" href="unialg_2.html#E3:35_1"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E7:35_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o0</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">B</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="E9:35_1"/><i><font color="Green" title="E24">A4</font></i>: 
( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</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="Maroon" title="c8">B</font> &amp; <font color="Maroon" title="c8">B</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="E10:35_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E7:35_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="unialg_2.html#T10" target="_self" title="UNIALG_2:th.10">Th10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">o1</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c2">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">n</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c2">U1</font> <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:35_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c9">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c8">B</font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E7:35_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="unialg_2.html#E9:35_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="unialg_2.html#E10:35_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="unialg_2.html#D7" target="_self" title="UNIALG_2:def.7">Def7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:35_1"/><i><font color="Green" title="E27">A7</font></i>: 
( <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">o0</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">o0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="txt" href="unialg_2.html#E5:35_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E15:35_1"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c6">o0</font>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E9:35_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="unialg_2.html#D5" target="_self" title="UNIALG_2:def.5">Def5</a></i>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">o0</font> = 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E5:35_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="unialg_2.html#T2" target="_self" title="UNIALG_2:th.2">Th2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">U0</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112" title="FINSEQ_2:th.112">FINSEQ_2:112</a></i>
;<br/>
<a NAME="E17:35_1"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c8">B</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E14:35_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<a NAME="E18:35_1"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c8">B</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T112" title="FINSEQ_2:th.112">FINSEQ_2:112</a></i>;<br/>
<a NAME="E19:35_1"/><b>then </b>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">o0</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E5:35_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="unialg_2.html#E14:35_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E20:35_1"/><b>then </b><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E21:35_1"/><b>then </b><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">o0</font> <a href="unialg_2.html#K2" title="UNIALG_2:func.2">/.</a> <font color="Maroon" title="c8">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="unialg_2.html#E15:35_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></i>;<br/>
<i><font color="Green" title="E31">A11</font></i>: <font color="Maroon" title="c9">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">o0</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">o0</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c8">B</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E12:35_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="unialg_2.html#E15:35_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="unialg_2.html#D6" target="_self" title="UNIALG_2:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="unialg_2.html#E14:35_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="unialg_2.html#E20:35_1"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E23:35_1"/>
( <font color="Maroon" title="c9">o1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">o1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">o1</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> )
 
<b>by </b><i><a class="txt" href="unialg_2.html#E12:35_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="unialg_2.html#E21:35_1"><i><font color="Green" title="E30">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E24:35_1"/>
<font color="Maroon" title="c4">x</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">U1</font>
 <b>by </b><i><a class="txt" href="unialg_2.html#E3:35_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="unialg_2.html#E22:35_1"><i><font color="Green" title="E31">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35"/>
 <a href="unialg_2.html#K6" title="UNIALG_2:func.6">Constants</a> <font color="Maroon" title="c1">U0</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">U1</font>
 ;<br/>


</div>
