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<b>let </b><font color="Maroon" title="c1">A</font> be  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a> <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>  <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>assume </b><a NAME="E1:44"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="pua2mss1.html#K6" title="PUA2MSS1:func.6">DomRel</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,$1 <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="pua2mss1.html#K6" title="PUA2MSS1:func.6">DomRel</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,$1 is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,$1 is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> &amp; <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,$1 is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> );<br/>
<a NAME="E2:44"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="txt" href="pua2mss1.html#E1:44"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="pua2mss1.html#T17" target="_self" title="PUA2MSS1:th.17">Th17</a></i>;<br/>
<div><i><font color="Green" title="E28">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:44_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font>]
 ;<br/><a NAME="E2:44_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="pua2mss1.html#K7" title="PUA2MSS1:func.7">|^</a> <font color="Maroon" title="c1">A</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">i</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">i</font></span>)</span> <a href="pua2mss1.html#K7" title="PUA2MSS1:func.7">|^</a> <font color="Maroon" title="c1">A</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="pua2mss1.html#T18" target="_self" title="PUA2MSS1:th.18">Th18</a>, <a class="ref" href="pua2mss1.html#T21" target="_self" title="PUA2MSS1:th.21">Th21</a></i>;<br/><b>hence </b><a NAME="E3:44_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="pua2mss1.html#E1:44_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="pua2mss1.html#T20" target="_self" title="PUA2MSS1:th.20">Th20</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:44"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="pua2mss1.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="pua2mss1.html#E3:44"><i><font color="Green" title="E28">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:44"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> ( <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">i</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp; <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">i</font> is <a href="relat_2.html#V3" title="RELAT_2:attr.3">symmetric</a> &amp; <font color="Maroon" title="c2">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">i</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> )
 ;<br/>


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