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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">[</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">S</font>, the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Olive" title="b1">S</font>, the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Olive" title="b1">S</font></span><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">]</a></span> );<br/>
<a NAME="E1:22_1_1"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font>] holds <br/><font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">z</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>] implies <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">z</font>] )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">S1</font>, <font color="Maroon" title="c6">S2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1_1_1"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c5">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E4:22_1_1_1"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">[</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S1</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S1</font>, the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c5">S1</font>, the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c5">S1</font></span><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:22_1_1_1"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<font color="Maroon" title="c6">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">and </span><br/><a NAME="E6:22_1_1_1"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">[</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">S2</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">S2</font>, the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c6">S2</font>, the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c6">S2</font></span><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:22_1_1_1"/>
( <font color="Maroon" title="c6">S2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  implies <font color="Maroon" title="c6">S2</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  )
 ;<br/>
<a NAME="E8:22_1_1_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
 the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c5">S1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c6">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E6:22_1_1_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="xtuple_0.html#T5" onmouseover="rs('xtuple_0/T5')" onmouseout="rh()">XTUPLE_0:5</a></span>;<br/>
<a NAME="E9:22_1_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">S2</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c6">S2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E6:22_1_1_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="xtuple_0.html#T5" onmouseover="rs('xtuple_0/T5')" onmouseout="rh()">XTUPLE_0:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:22_1_1_1"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1_1"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E4:22_1_1_1"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E5:22_1_1_1"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="txt" href="#E6:22_1_1_1"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="txt" href="#E8:22_1_1_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="xtuple_0.html#T5" onmouseover="rs('xtuple_0/T5')" onmouseout="rh()">XTUPLE_0:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c2">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1_1"/><span class="lab"><font color="Green" title="E14">A7</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="tarski.html#S1" onmouseover="rs('tarski/S1')" onmouseout="rh()">TARSKI:sch 1</a>(<a class="txt" href="#E1:22_1_1"><span class="lab"><font color="Green" title="E13">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c2">X</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:22_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff  ex <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:22_1_1_2"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> implies  ex <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> ) implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="22_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:22_1_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:22_1_1_2_1"/><span class="lab"><font color="Green" title="E15">A8</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:22_1_1_2_1"/><span class="lab"><font color="Green" title="E16">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <span class="kw"> such that </span><br/><a NAME="E6:22_1_1_2_1"/><span class="lab"><font color="Green" title="E17">A10</font></span>: 
<font color="Maroon" title="c5">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">and </span><br/><a NAME="E7:22_1_1_2_1"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">[</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S</font>, the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c5">S</font>, the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c5">S</font></span><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1_2_1"><span class="lab"><font color="Green" title="E16">A9</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">S</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )</span><br/>

<a NAME="E8:22_1_1_2_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1_2_1"><span class="lab"><font color="Green" title="E15">A8</font></span></a>, <a class="txt" href="#E4:22_1_1_2_1"><span class="lab"><font color="Green" title="E16">A9</font></span></a>, <a class="txt" href="#E6:22_1_1_2_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a>, <a class="ref" href="mcart_1.html#T80" onmouseover="rs('mcart_1/T80')" onmouseout="rh()">MCART_1:80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:22_1_1_2_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:22_1_1_2_1"><span class="lab"><font color="Green" title="E17">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">given </span><font color="Maroon" title="c4">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <span class="kw"> such that </span><a NAME="E3:22_1_1_2"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font>
 <span class="kw">and </span><br/><a NAME="E4:22_1_1_2"/><span class="lab"><font color="Green" title="E16">A12</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E5:22_1_1_2"/><span class="lab"><font color="Green" title="E17">A13</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<a NAME="E6:22_1_1_2"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">S</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1_2"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E7:22_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A14</font></span>: 
 the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="ref" href="partfun1.html#D3" onmouseover="rs('partfun1/D3')" onmouseout="rh()">PARTFUN1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">C</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1_2"><span class="lab"><font color="Green" title="E16">A12</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E10:22_1_1_2"/><span class="lab"><font color="Green" title="E19">A15</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">[</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">S</font>, the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S</font>, the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S</font></span><a href="xtuple_0.html#K6" title="XTUPLE_0:func.6">]</a></span>
 ;<br/>
<a NAME="E11:22_1_1_2"/>
<font color="Maroon" title="c5">C</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msuhom_1.html#T2" onmouseover="rs('msuhom_1/T2')" onmouseout="rh()">MSUHOM_1:2</a></span>;<br/>
<a NAME="E12:22_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A16</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
;<br/>
<a NAME="E13:22_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:22_1_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a></span>;<br/>
<a NAME="E14:22_1_1_2"/><span class="kw">then </span>
 the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:22_1_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="ref" href="partfun1.html#D3" onmouseover="rs('partfun1/D3')" onmouseout="rh()">PARTFUN1:def 3</a></span>;<br/>
<a NAME="E15:22_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>)</span>)</span>,<span class="p2">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> (<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c1">A</font>)</span>)</span></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:22_1_1_2"><span class="lab"><font color="Green" title="E16">A12</font></span></a>, <a class="txt" href="#E5:22_1_1_2"><span class="lab"><font color="Green" title="E17">A13</font></span></a>, <a class="txt" href="#E10:22_1_1_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a>, <a class="txt" href="#E7:22_1_1_2"><span class="lab"><font color="Green" title="E18">A14</font></span></a>, <a class="ref" href="mcart_1.html#T80" onmouseover="rs('mcart_1/T80')" onmouseout="rh()">MCART_1:80</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:22_1_1_2"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22_1_1"><span class="lab"><font color="Green" title="E14">A7</font></span></a>, <a class="txt" href="#E3:22_1_1_2"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="txt" href="#E10:22_1_1_2"><span class="lab"><font color="Green" title="E19">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
