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<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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> <br/> for <font color="Olive" title="b3">R</font> being  <a href="relset_1.html#NM2" title="RELSET_1:NM.2">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> holds  <font color="Olive" title="b3">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="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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></span>)</span> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b2">j</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="relset_1.html#NM2" title="RELSET_1:NM.2">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> holds  <font color="Olive" title="b1">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="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">j</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">R</font> be   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">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="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">j</font></span><br/>





<span class="kw">defpred </span><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> ] <span class="kw">means </span><font color="Maroon" title="c4">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="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,$1;<br/>
<a NAME="E1:41"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#T17" target="_self" title="PUA2MSS1:th.17">Th17</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E24">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">j</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><font color="Maroon" title="c4">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="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c4">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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="pua2mss1.html#K7" title="PUA2MSS1:func.7">|^</a> <font color="Maroon" title="c1">A</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#T18" target="_self" title="PUA2MSS1:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">i</font></span>)</span> <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="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="pua2mss1.html#T18" target="_self" title="PUA2MSS1:th.18">Th18</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:41_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E3:41"/>
for <font color="Olive" title="b1">j</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">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E24">A2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E4:41"/>
<font color="Maroon" title="c4">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="c2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">R</font> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="pua2mss1.html#K8" title="PUA2MSS1:func.8">|^</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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