<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E24</font></i>: 
 <a href="msafree2.html#K3">InnerVertices</a> <font color="Maroon">S1</font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K3">InnerVertices</a> <font color="Maroon">S2</font>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msafree2.html#D6">MSAFREE2:def 6</a><br/>

<b>assume </b><a NAME="E2:21"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="circcomb.html#K3" target="_self">+*</a> <font color="Maroon">S2</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font>;<br/>
<b>set </b><font color="Maroon">r1</font> = the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font>;<br/>
<b>set </b><font color="Maroon">r2</font> = the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font>;<br/>
<b>let </b><font color="Maroon">o1</font> be   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">S</font>, <font color="Maroon">o2</font> be   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E3:21"/><i><font color="Green">E38</font></i>: 
 <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a>  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">o2</font>
 ;<br/>

<a NAME="E4:21"/><i><font color="Green">E42</font></i>: 
( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_4.html#K1">+*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:21"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> &amp; the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> <a href="xboole_0.html#K2">\/</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:21"/><b>then </b>
( ( ( not <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> ) or <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> ) &amp; ( ( not <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> ) or <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> ) )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E7:21"/><b>then </b><i><font color="Green">E45</font></i>: 
( ( ( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> ) or ( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> ) ) &amp; ( ( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> ) or ( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> ) ) )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1"/>
( ( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> ) or ( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> ) )
 </span><b>by </b><i>, , , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><b>suppose </b></a><a NAME="E1:21_1_1"/><i><font color="Green">E75</font></i>: 
( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S1</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S1</font> as  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  <b>by </b><i><a class="ref" href="msualg_1.html#D5">MSUALG_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">o1</font>, <font color="Maroon">p2</font> = <font color="Maroon">o2</font> as   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E4:21_1_1"/>
(  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font> )
 ;<br/><b>hence </b><a NAME="E5:21_1_1"/>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i>, , <a class="ref" href="msafree2.html#D6">MSAFREE2:def 6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_2"><b>suppose </b></a><a NAME="E1:21_1_2"/><i><font color="Green">E78</font></i>: 
( the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">o1</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> &amp; the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">o2</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">S2</font> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S2</font> as  non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  <b>by </b><i><a class="ref" href="msualg_1.html#D5">MSUALG_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">o1</font>, <font color="Maroon">p2</font> = <font color="Maroon">o2</font> as   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/><a NAME="E4:21_1_2"/>
(  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> &amp;  <a href="msualg_1.html#K2">the_result_sort_of</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">S2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font> )
 ;<br/><b>hence </b><a NAME="E5:21_1_2"/>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i>, , <a class="ref" href="msafree2.html#D6">MSAFREE2:def 6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
