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<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b3">A</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Olive" title="b2">S</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Olive" title="b4">o</font></span>)</span> <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>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b4">o</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</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">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Olive" title="b1">S</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Olive" title="b3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b3">o</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <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">A</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">S</font><br/> for <font color="Olive" title="b2">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c3">S</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Olive" title="b2">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b2">o</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">A</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">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="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c3">S</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Olive" title="b1">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font></span>)</span></span><br/>









<b>set </b><font color="Maroon" title="c6">O</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font>;<br/>
<b>set </b><font color="Maroon" title="c7">f</font> =  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span>;<br/>
<a NAME="E1:27"/><i><font color="Green" title="E13">A1</font></i>: 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="msualg_1.html#D5" title="MSUALG_1:def.5">MSUALG_1:def 5</a></i>;<br/>
<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T113" title="ZFMISC_1:th.113">ZFMISC_1:113</a></i>;<br/>
<a NAME="E3:27"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c3">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T12" target="_self" title="PRALG_2:th.12">Th12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:27"/><i><font color="Green" title="E16">A4</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <span class="p2">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p3">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">o</font> ) )
 <b>by </b><i><a class="txt" href="pralg_2.html#E1:27"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="pralg_2.html#E2:27"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="funct_5.html#T39" title="FUNCT_5:th.39">FUNCT_5:39</a></i>;<br/>
<a NAME="E6:27"/><i><font color="Green" title="E17">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">o</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="pralg_2.html#E1:27"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="pralg_2.html#E3:27"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:27"/><i><font color="Green" title="E18">A6</font></i>: 
( <font color="Maroon" title="c2">i</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="c8">g</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">o</font> )
 
<b>by </b><i><a class="txt" href="pralg_2.html#E5:27"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E8:27"/><i><font color="Green" title="E19">A7</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">o</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">o</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#D1" title="MCART_1:def.1">MCART_1:def 1</a>, <a class="ref" href="mcart_1.html#D2" title="MCART_1:def.2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:27"/>
 ex <font color="Olive" title="b1">U0</font> being   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c3">S</font> st <br/>( <font color="Olive" title="b1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K11" title="PRALG_2:func.11">OPER</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <font color="Olive" title="b1">U0</font> )
 
<b>by </b><i><a class="ref" href="pralg_2.html#D18" target="_self" title="PRALG_2:def.18">Def18</a></i>;<br/>
<b>then </b><font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> = 
the <a href="msualg_1.html#U5" title="MSUALG_1:sel.5">Charact</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font>
<b>by </b><i><a class="txt" href="pralg_2.html#E6:27"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="pralg_2.html#E7:27"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="pralg_2.html#E8:27"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="funct_5.html#D4" title="FUNCT_5:def.4">FUNCT_5:def 4</a></i>
<br/>.= 
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font></span>)</span>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E11:27"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K12" title="PRALG_2:func.12">?.</a> <font color="Maroon" title="c5">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c5">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c2">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="pralg_2.html#E5:27"><i><font color="Green" title="E16">A4</font></i></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>
