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<span class="kw">let </span><font color="Maroon" title="c1">D</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">p3</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> holds   <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,1,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Olive" title="b2">p2</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">p3</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,1,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>;<br/>
<a NAME="E1:32"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E2:32"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</font></span>: <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c2">p1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
;<br/>
<a NAME="E4:32"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
<span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c3">p2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>
;<br/>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,1,2 = 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1">(<span class="default"><a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,1,<span class="p2">(<span class="default"><span class="p3"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span></span>)</span>,2,<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="finseq_7.html#D2" target="_self" title="FINSEQ_7:def.2">Def2</a></span>
<br/>.= 
 <a href="finseq_7.html#K1" title="FINSEQ_7:func.1">Replace</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,2,<span class="p1">(<span class="default"><span class="p2"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="finseq_7.html#T17" target="_self" title="FINSEQ_7:th.17">Th17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:32"/>
 <a href="finseq_7.html#K2" title="FINSEQ_7:func.2">Swap</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>,1,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c3">p2</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c4">p3</font></span><a href="finseq_4.html#K3" title="FINSEQ_4:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="ref" href="finseq_7.html#T18" target="_self" title="FINSEQ_7:th.18">Th18</a></span>; <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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