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<span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="exchsort.html#NM1" title="EXCHSORT:NM.1">array</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">B</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="c1">A</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>) implies <font color="Maroon" title="c4">B</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c4">B</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="c1">A</font>,<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">B</font> is  <a href="valued_0.html#V3" title="VALUED_0:attr.3">real-valued</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="valued_0.html#D9" title="VALUED_0:def.9">VALUED_0:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c4">B</font> or <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:65_1_1"/>
<font color="Maroon" title="c5">z</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="c4">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> </span><br/>

<a NAME="E3:65_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c5">z</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="c1">A</font> &amp; ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> ) &amp; ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="funct_7.html#T99" title="FUNCT_7:th.99">FUNCT_7:99</a></span>;<br/>
<a NAME="E4:65_1_1"/>
( (  not <font color="Maroon" title="c2">x</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="c1">A</font> or  not <font color="Maroon" title="c3">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="c1">A</font> ) implies <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="funct_7.html#D12" title="FUNCT_7:def.12">FUNCT_7:def 12</a></span>;<br/>
<a NAME="E5:65_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font> or <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:65_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="exchsort.html#T29" target="_self" title="EXCHSORT:th.29">TSa</a>, <a class="ref" href="exchsort.html#T31" target="_self" title="EXCHSORT:th.31">TSb</a>, <a class="ref" href="exchsort.html#T33" target="_self" title="EXCHSORT:th.33">TSc</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:65_1_1"/>
<font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> is  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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