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<div class="add">

<b>set </b><font color="Maroon">S</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">SORTS</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> ;<br/>
<b>consider </b><font color="Maroon">CHARACT</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:73_1"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">o</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="finseq_2.html#K3">*</a>   st <font color="Olive">o</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> holds <br/><font color="Maroon">CHARACT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">o</font>,<font color="Olive">p</font>)
 <b>from </b><i><a class="ref" href="circcomb.html#S1" target="_self">CIRCCOMB:sch 1</a>();<br/></i>
<a NAME="E3:73_1"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">CHARACT</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:73_1"/>
<font color="Maroon">CHARACT</font> is <a href="funcop_1.html#V1">Function-yielding</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funcop_1.html#D6">FUNCOP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:73_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">CHARACT</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">x</font> as   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="circcomb.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<a NAME="E4:73_1_1"/>
<font color="Maroon">CHARACT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">o</font>,<font color="Maroon">p</font>)
 
<b>by </b><i><a class="ref" href="circcomb.html#T4" target="_self">Th4</a>, , <a class="ref" href="circcomb.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E5:73_1_1"/>
<font color="Maroon">CHARACT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> is    <a href="hidden.html#M1">set</a> 
 <b>by </b><i>, , <a class="ref" href="circcomb.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CHARACT</font> = <font color="Maroon">CHARACT</font> as   <a href="pboole.html#NM2">ManySortedFunction</a> of the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:73_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">o</font> = <font color="Maroon">i</font> as   <a href="circcomb.html#NM1">Gate</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/><b>reconsider </b><font color="Maroon">p</font> = the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> as    <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="finseq_2.html#K3">*</a>  <b>by </b><i><a class="ref" href="circcomb.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon">v</font> = the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="circcomb.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E5:73_1_2"/>
<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K6">BOOLEAN</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E6:73_1_2"/><b>then </b>
( <font color="Maroon">CHARACT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">o</font>,<font color="Maroon">p</font>) &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="margrel1.html#K6">BOOLEAN</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K6">BOOLEAN</a>  )
 <b>by </b><i><a class="ref" href="circcomb.html#T4" target="_self">Th4</a>, , <a class="ref" href="circcomb.html#T5" target="_self">Th5</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><b>hence </b><a NAME="E7:73_1_2"/>
<font color="Maroon">CHARACT</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="circcomb.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">CHARACT</font> = <font color="Maroon">CHARACT</font> as    <a href="pboole.html#M3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span> <a href="pboole.html#K7">*</a> the <a href="msualg_1.html#U3">ResultSort</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/>
<b>take </b>
 <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span>,<font color="Maroon">CHARACT</font> #)
;<br/>

<b>thus </b><a NAME="E8:73_1"/>
( the <a href="msualg_1.html#U4">Sorts</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span>,<font color="Maroon">CHARACT</font> #) <a href="pboole.html#R6">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a>  &amp; ( for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">p</font> being   <a href="finseq_2.html#M2">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="finseq_2.html#K3">*</a>   st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U1">OperSymbols</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> the <a href="msualg_1.html#U2">Arity</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">g</font> holds <br/>the <a href="msualg_1.html#U5">Charact</a> of <a href="msualg_1.html#G3">MSAlgebra</a>(# <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="pre_circ.html#K2">--&gt;</a> <a href="margrel1.html#K6">BOOLEAN</a> </span>)</span>,<font color="Maroon">CHARACT</font> #) <a href="funct_1.html#K1">.</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">g</font>,<font color="Olive">p</font>) ) )
 <b>by </b><i><a class="ref" href="circcomb.html#T4" target="_self">Th4</a></i>;<br/>


</div>
