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<b>let </b><font color="Maroon" title="c5">IT</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> implies ( <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> ) ) &amp; ( (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> or  not <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> ) implies ( <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> ) ) )</span><br/>

<b>thus </b><a NAME="E1:23_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> implies ( <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( (  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> or  not <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> ) implies ( <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_2_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> )</span><br/>

<a NAME="E2:23_1_2_1"/><b>then </b><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c3">i</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> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">j</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> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E3:23_1_2_1"/><b>then </b>
( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">j</font> )
 
<b>by </b><i><a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_2_1"/>
( <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span>,<font color="Maroon" title="c4">j</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="finseq_7.html#E2:23_1_2_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="funct_7.html#D12" title="FUNCT_7:def.12">FUNCT_7:def 12</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>assume </b><a NAME="E2:23_1_2"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font> or  not <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</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">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<a NAME="E3:23_1_2"/><b>then </b>
(  not <font color="Maroon" title="c3">i</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> <font color="Maroon" title="c2">f</font> or  not <font color="Maroon" title="c4">j</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> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1_2"/>
( <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_7.html#K10" title="FUNCT_7:func.10">Swap</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> iff <font color="Maroon" title="c5">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="ref" href="funct_7.html#D12" title="FUNCT_7:def.12">FUNCT_7:def 12</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>
