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<b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="osafree.html#K5" title="OSAFREE:func.5">ParsedTerms</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><a href="osafree.html#K5" title="OSAFREE:func.5">ParsedTerms</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">A</font> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Olive" title="b1">o</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> ) &amp; ( for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c4">B</font> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Olive" title="b1">o</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font> ) implies <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_1_2"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c3">A</font> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Olive" title="b1">o</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E2:22_1_2"/><i><font color="Green" title="E13">A4</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c4">B</font> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Olive" title="b1">o</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font></span><br/>

<a NAME="E3:22_1_2"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<b>assume </b><a NAME="E1:22_1_2_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">i</font> as   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:22_1_2_1"/>
( <font color="Maroon" title="c3">A</font> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c6">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">B</font> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c6">s</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="#E1:22_1_2"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="#E2:22_1_2"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:22_1_2_1"/>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ; <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>hence </b><a NAME="E4:22_1_2"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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