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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="pua2mss1.html#M3" title="PUA2MSS1:mode.3">a_partition</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" title="PUA2MSS1:func.9">LimDomRel</a> <font color="Maroon" title="c1">A</font></span>)</span> holds <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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>   st <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R7" title="PUA2MSS1:pred.7">can_be_characterized_by</a> <font color="Olive" title="b2">S</font> holds <br/> <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Olive" title="b1">P</font> <a href="pua2mss1.html#R5" title="PUA2MSS1:pred.5">is_rougher_than</a> <font color="Olive" title="b2">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">P</font> be    <a href="pua2mss1.html#M3" title="PUA2MSS1:mode.3">a_partition</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" title="PUA2MSS1:func.9">LimDomRel</a> <font color="Maroon" title="c1">A</font></span>)</span> implies 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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>   st <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R7" title="PUA2MSS1:pred.7">can_be_characterized_by</a> <font color="Olive" title="b1">S</font> holds <br/> <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font> <a href="pua2mss1.html#R5" title="PUA2MSS1:pred.5">is_rougher_than</a> <font color="Olive" title="b1">S</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:72"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
<font color="Maroon" title="c2">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K9" title="PUA2MSS1:func.9">LimDomRel</a> <font color="Maroon" title="c1">A</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>   st <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R7" title="PUA2MSS1:pred.7">can_be_characterized_by</a> <font color="Olive" title="b1">S</font> holds <br/> <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font> <a href="pua2mss1.html#R5" title="PUA2MSS1:pred.5">is_rougher_than</a> <font color="Olive" title="b1">S</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">S1</font> =  <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">S2</font> be  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#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R7" title="PUA2MSS1:pred.7">can_be_characterized_by</a> <font color="Maroon" title="c4">S2</font> implies  <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font> <a href="pua2mss1.html#R5" title="PUA2MSS1:pred.5">is_rougher_than</a> <font color="Maroon" title="c4">S2</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">G</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c4">S2</font>, <font color="Maroon" title="c6">Q</font> being    <a href="pua2mss1.html#M2" title="PUA2MSS1:mode.2">IndexedPartition</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><a NAME="E3:72"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R6" title="PUA2MSS1:pred.6">can_be_characterized_by</a> <font color="Maroon" title="c4">S2</font>,<font color="Maroon" title="c5">G</font>,<font color="Maroon" title="c6">Q</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pua2mss1.html#D17" target="_self" title="PUA2MSS1:def.17">PUA2MSS1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font> <a href="pua2mss1.html#R5" title="PUA2MSS1:pred.5">is_rougher_than</a> <font color="Maroon" title="c4">S2</font></span><br/>

<a NAME="E4:72"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c5">G</font> is    <a href="pua2mss1.html#M4" title="PUA2MSS1:mode.4">IndexedPartition</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></span>;<br/>
<a NAME="E5:72"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c5">G</font> as   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c4">S2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="msualg_1.html#D8" title="MSUALG_1:def.8">MSUALG_1:def 8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">R</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> as    <a href="pua2mss1.html#M4" title="PUA2MSS1:mode.4">IndexedPartition</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></span>;<br/>
<a NAME="E8:72"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">R</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="c4">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">SG</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S2</font>,<span class="p1">(<span class="default"><a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<a NAME="E10:72"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o</font> is   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>, <font color="Olive" title="b2">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b2">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c1">A</font></span>)</span> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<a NAME="E11:72"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>$1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $2;<br/>
<a NAME="E12:72"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="pua2mss1.html#T28" target="_self" title="PUA2MSS1:th.28">Th28</a></span>;<br/>
<a NAME="E13:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A9</font></span>: 
for <font color="Olive" title="b1">r</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> holds <br/> ex <font color="Olive" title="b2">p</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">p</font>] )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">em</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E15:72"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">em</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">em</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font> )
 <span class="kw">and </span><br/><a NAME="E16:72"/><span class="lab"><font color="Green" title="E51">A11</font></span>: 
for <font color="Olive" title="b1">r</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>,<font color="Maroon" title="c10">em</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="#E13:72"><span class="lab"><font color="Green" title="E49">A9</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">em</font> = <font color="Maroon" title="c10">em</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font></span>)</span>,<font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:72"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<a NAME="E18:72"/><span class="lab"><font color="Green" title="E52">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">em</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:72"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E19:72"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">em</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c11">em</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:72"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">em</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">R</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S2</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E18:72"><span class="lab"><font color="Green" title="E52">A12</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pua2mss1.html#D14" target="_self" title="PUA2MSS1:def.14">PUA2MSS1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c12">f</font>,<font color="Olive" title="b1">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c4">S2</font>, <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font> ex <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="c4">S2</font> st <br/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> $1</span>)</span>,<font color="Olive" title="b1">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b2">o</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">p</font> );<br/>
<a NAME="E21:72"/><span class="lab"><font color="Green" title="E53">A13</font></span>: 
for <font color="Olive" title="b1">s2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s2</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="c4">S2</font> holds <br/> ex <font color="Olive" title="b2">s1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">s1</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s2</font>,<font color="Olive" title="b2">s1</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">s2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">s2</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="c4">S2</font> implies  ex <font color="Olive" title="b1">s1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s1</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">s2</font>,<font color="Olive" title="b1">s1</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_1"/>
<font color="Maroon" title="c13">s2</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="c4">S2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">s1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s1</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">s2</font>,<font color="Olive" title="b1">s1</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">s2</font> = <font color="Maroon" title="c13">s2</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font> ;<br/>
<a NAME="E3:72_1"/>
<font color="Maroon" title="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">s2</font></span>)</span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="pua2mss1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font><span class="kw"> such that </span><br/><a NAME="E5:72_1"/><span class="lab"><font color="Green" title="E54">A14</font></span>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">s2</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c15">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="pua2mss1.html#T4" target="_self" title="PUA2MSS1:th.4">Th4</a>, <a class="ref" href="pua2mss1.html#T28" target="_self" title="PUA2MSS1:th.28">Th28</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">p</font> = <font color="Maroon" title="c15">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</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="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">take </span><font color="Maroon" title="c17">s1</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c14">s2</font></span>)</span>,<font color="Maroon" title="c16">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">s1</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">s2</font>,<font color="Maroon" title="c17">s1</font>] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">o</font> = <font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c14">s2</font> as   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font>;<br/>
<a NAME="E9:72_1"/><span class="lab"><font color="Green" title="E55">A15</font></span>: 
<font color="Maroon" title="c19">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font>
 
;<br/>
<a NAME="E10:72_1"/><span class="lab"><font color="Green" title="E56">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:72_1"/><span class="lab"><font color="Green" title="E57">A17</font></span>: 
 <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">S2</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">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E12:72_1"/><span class="lab"><font color="Green" title="E58">A18</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</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">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E13:72_1"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span> is    <a href="pua2mss1.html#M2" title="PUA2MSS1:mode.2">IndexedPartition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></span>;<br/>
<a NAME="E14:72_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D3" target="_self" title="PUA2MSS1:def.3">Def3</a></span>;<br/>
<a NAME="E15:72_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A19</font></span>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/>
<a NAME="E16:72_1"/>
<font color="Maroon" title="c14">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></span>;<br/>
<a NAME="E17:72_1"/><span class="kw">then </span>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:72_1"><span class="lab"><font color="Green" title="E58">A18</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/>
<a NAME="E18:72_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:72_1"><span class="lab"><font color="Green" title="E59">A19</font></span></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E61">A21</font></span>:  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72_1"><span class="lab"><font color="Green" title="E57">A17</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c14">s2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></span>
;<br/>
<a NAME="E20:72_1"/><span class="kw">then </span>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E9:72_1"><span class="lab"><font color="Green" title="E55">A15</font></span></a>, <a class="txt" href="#E10:72_1"><span class="lab"><font color="Green" title="E56">A16</font></span></a>, <a class="txt" href="#E18:72_1"><span class="lab"><font color="Green" title="E60">A20</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:72_1"/>
<font color="Maroon" title="c17">s1</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E46">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">s2</font>,<font color="Maroon" title="c17">s1</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c16">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <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="c4">S2</font> st <br/>( <font color="Maroon" title="c17">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c13">s2</font></span>)</span>,<font color="Maroon" title="c16">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">p</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c14">s2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c13">s2</font></span>)</span>,<font color="Maroon" title="c16">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s2</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">p</font> )</span><br/>

<span class="kw">thus </span><a NAME="E22:72_1"/>
( <font color="Maroon" title="c17">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c13">s2</font></span>)</span>,<font color="Maroon" title="c16">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">s2</font> &amp;  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c14">s2</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c16">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72_1"><span class="lab"><font color="Green" title="E54">A14</font></span></a>, <a class="txt" href="#E19:72_1"><span class="lab"><font color="Green" title="E61">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c13">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E23:72"/><span class="lab"><font color="Green" title="E54">A22</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">g</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">S2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp; ( for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</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="c4">S2</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Maroon" title="c13">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="#E21:72"><span class="lab"><font color="Green" title="E53">A13</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c4">S2</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:72"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">g</font> <a href="pua2mss1.html#R3" title="PUA2MSS1:pred.3">form_morphism_between</a> <font color="Maroon" title="c4">S2</font>, <a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E25:72"/><span class="lab"><font color="Green" title="E55">A23</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f</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="c4">S2</font> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">g</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">S2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="pua2mss1.html#D13" target="_self" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">g</font> &amp; ( for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">p</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">o</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="c4">S2</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> holds <br/><font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font></span>)</span> ) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">c</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span><br/><span class="kw">set </span><font color="Maroon" title="c16">s</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font>;<br/><a NAME="E1:72_2"/><span class="lab"><font color="Green" title="E56">A24</font></span>: 
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/><a NAME="E2:72_2"/><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/><a NAME="E3:72_2"/>
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E4:72_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">em</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:72"><span class="lab"><font color="Green" title="E51">A11</font></span></a></span>;<br/><a NAME="E5:72_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A26</font></span>: 
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c18">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E7:72_2"/><span class="lab"><font color="Green" title="E59">A27</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c15">c</font></span>)</span>,<font color="Maroon" title="c17">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:72_2"/><span class="lab"><font color="Green" title="E60">A28</font></span>: 
<font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">o</font>
 <span class="kw">and </span><br/><a NAME="E9:72_2"/><span class="lab"><font color="Green" title="E61">A29</font></span>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:72"><span class="lab"><font color="Green" title="E54">A22</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c17">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</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="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">s2</font> = <font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c15">c</font> as   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font>;<br/><a NAME="E13:72_2"/>
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span> is    <a href="pua2mss1.html#M2" title="PUA2MSS1:mode.2">IndexedPartition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="pua2mss1.html#D16" target="_self" title="PUA2MSS1:def.16">Def16</a></span>;<br/><a NAME="E14:72_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A30</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D3" target="_self" title="PUA2MSS1:def.3">Def3</a></span>;<br/><a NAME="E15:72_2"/><span class="lab"><font color="Green" title="E63">A31</font></span>: 
<font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72_2"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></span>;<br/><a NAME="E16:72_2"/><span class="lab"><font color="Green" title="E64">A32</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</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">S2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E17:72_2"/><span class="lab"><font color="Green" title="E65">A33</font></span>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span> is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72_2"><span class="lab"><font color="Green" title="E62">A30</font></span></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></span>;<br/><a NAME="E18:72_2"/><span class="lab"><font color="Green" title="E66">A34</font></span>: 
the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U4" title="MSUALG_1:sel.4">Charact</a> of <font color="Maroon" title="c7">G</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:72_2"><span class="lab"><font color="Green" title="E63">A31</font></span></a>, <a class="txt" href="#E16:72_2"><span class="lab"><font color="Green" title="E64">A32</font></span></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></span>;<br/><a NAME="E19:72_2"/><span class="lab"><font color="Green" title="E67">A35</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E20:72_2"/><span class="lab"><font color="Green" title="E68">A36</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:72_2"><span class="lab"><font color="Green" title="E65">A33</font></span></a>, <a class="txt" href="#E18:72_2"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="ref" href="relat_1.html#T25" title="RELAT_1:th.25">RELAT_1:25</a></span>;<br/><a NAME="E21:72_2"/>
<font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c18">o</font>,<font color="Maroon" title="c7">G</font>
 ;<br/><a NAME="E22:72_2"/><span class="kw">then </span>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:72_2"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="txt" href="#E19:72_2"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="txt" href="#E20:72_2"><span class="lab"><font color="Green" title="E68">A36</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><a NAME="E23:72_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A37</font></span>: 
<span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72_2"><span class="lab"><font color="Green" title="E59">A27</font></span></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/><a NAME="E24:72_2"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72_2"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="#E19:72_2"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></span>;<br/><a NAME="E25:72_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A38</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72_2"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="#E9:72_2"><span class="lab"><font color="Green" title="E61">A29</font></span></a>, <a class="ref" href="relat_1.html#T156" title="RELAT_1:th.156">RELAT_1:156</a></span>;<br/><a NAME="E26:72_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72_2"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="#E17:72_2"><span class="lab"><font color="Green" title="E65">A33</font></span></a>, <a class="txt" href="#E18:72_2"><span class="lab"><font color="Green" title="E66">A34</font></span></a>, <a class="ref" href="relat_1.html#T157" title="RELAT_1:th.157">RELAT_1:157</a></span>;<br/><a NAME="E27:72_2"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c20">s2</font>,<font color="Maroon" title="c1">A</font></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:72_2"><span class="lab"><font color="Green" title="E70">A38</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E28:72_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A39</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:72_2"><span class="lab"><font color="Green" title="E69">A37</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E29:72_2"/><span class="lab"><font color="Green" title="E72">A40</font></span>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72_2"><span class="lab"><font color="Green" title="E60">A28</font></span></a>, <a class="txt" href="#E19:72_2"><span class="lab"><font color="Green" title="E67">A35</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E30:72_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A41</font></span>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">c</font>
 ;<br/><a NAME="E31:72_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:72_2"><span class="lab"><font color="Green" title="E71">A39</font></span></a>, <a class="txt" href="#E29:72_2"><span class="lab"><font color="Green" title="E72">A40</font></span></a></span>;<br/><a NAME="E32:72_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c15">c</font>,<font color="Maroon" title="c7">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">aa</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/><span class="kw">hence </span><a NAME="E33:72_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p2">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E1:72_2"><span class="lab"><font color="Green" title="E56">A24</font></span></a>, <a class="txt" href="#E2:72_2"><span class="lab"><font color="Green" title="E57">A25</font></span></a>, <a class="txt" href="#E5:72_2"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="txt" href="#E30:72_2"><span class="lab"><font color="Green" title="E73">A41</font></span></a>, <a class="ref" href="pua2mss1.html#T3" target="_self" title="PUA2MSS1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E27:72"/>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">p</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">o</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="c4">S2</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> holds <br/><font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font></span>)</span> ) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c15">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Maroon" title="c15">o</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="c4">S2</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> holds <br/><font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c16">p</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">o</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="c4">S2</font> &amp; <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> implies <font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:72_3"/>
<font color="Maroon" title="c15">o</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="c4">S2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font> implies <font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">s</font> = <font color="Maroon" title="c15">o</font> as   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font> ;<br/><span class="kw">assume </span><a NAME="E3:72_3"/><span class="lab"><font color="Green" title="E56">A42</font></span>: 
<font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">q</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S2</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">r</font> = <font color="Maroon" title="c9">SG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c18">q</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="pua2mss1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">p9</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c21">o9</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E7:72_3"/><span class="lab"><font color="Green" title="E57">A43</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c20">p9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:72_3"/><span class="lab"><font color="Green" title="E58">A44</font></span>: 
<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o9</font>
 <span class="kw">and </span><br/><a NAME="E9:72_3"/><span class="lab"><font color="Green" title="E59">A45</font></span>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c20">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:72"><span class="lab"><font color="Green" title="E54">A22</font></span></a></span>;<br/><a NAME="E10:72_3"/>
<font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">o1</font> being   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c23">p1</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <span class="kw"> such that </span><br/><a NAME="E12:72_3"/><span class="lab"><font color="Green" title="E60">A46</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">o1</font>,<font color="Maroon" title="c23">p1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E13:72_3"/><span class="lab"><font color="Green" title="E61">A47</font></span>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c23">p1</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c22">o1</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E46">A6</font></span></a></span>;<br/><a NAME="E14:72_3"/>
<font color="Maroon" title="c20">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72_3"><span class="lab"><font color="Green" title="E57">A43</font></span></a>, <a class="txt" href="#E12:72_3"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/><a NAME="E15:72_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A48</font></span>: 
the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:72_3"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="txt" href="#E13:72_3"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/> <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o9</font>,<font color="Maroon" title="c7">G</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">q</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72_3"><span class="lab"><font color="Green" title="E58">A44</font></span></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c19">r</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></span>
;<br/><a NAME="E17:72_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">em</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">r</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:72"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E9:72_3"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="ref" href="pua2mss1.html#T5" target="_self" title="PUA2MSS1:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E18:72_3"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c16">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E15:72_3"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E29:72"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E30:72"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> or <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_4"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c16">s</font> = <font color="Maroon" title="c15">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c16">s</font>;<br/>
<a NAME="E4:72_4"/><span class="lab"><font color="Green" title="E56">A49</font></span>: 
<font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</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">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></span>;<br/>
<a NAME="E5:72_4"/><span class="lab"><font color="Green" title="E57">A50</font></span>: 
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></span>;<br/>
<a NAME="E6:72_4"/><span class="lab"><font color="Green" title="E58">A51</font></span>: 
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_4"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E7:72_4"/>
<font color="Maroon" title="c11">em</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_4"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/>
<a NAME="E8:72_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A52</font></span>: 
<font color="Maroon" title="c8">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:72"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E6:72_4"><span class="lab"><font color="Green" title="E58">A51</font></span></a></span>;<br/>
<a NAME="E9:72_4"/><span class="lab"><font color="Green" title="E60">A53</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">R</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c17">a</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E25:72"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="txt" href="#E4:72_4"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E10:72_4"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:72"><span class="lab"><font color="Green" title="E55">A23</font></span></a>, <a class="ref" href="pua2mss1.html#D15" target="_self" title="PUA2MSS1:def.15">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:72_4"/>
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72_4"><span class="lab"><font color="Green" title="E57">A50</font></span></a>, <a class="txt" href="#E8:72_4"><span class="lab"><font color="Green" title="E59">A52</font></span></a>, <a class="txt" href="#E9:72_4"><span class="lab"><font color="Green" title="E60">A53</font></span></a>, <a class="ref" href="pua2mss1.html#T3" target="_self" title="PUA2MSS1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E31:72"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">g</font></span><br/>

<span class="kw">thus </span><a NAME="E32:72"/>
the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">s1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">s1</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span> or <font color="Maroon" title="c15">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_5"/>
<font color="Maroon" title="c15">s1</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 <span class="p1">(<span class="default"><a href="pua2mss1.html#K11" title="PUA2MSS1:func.11">MSSign</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">P</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">o</font> being   <a href="pua2mss1.html#NM2" title="PUA2MSS1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c17">p</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">P</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <span class="kw"> such that </span><br/><a NAME="E3:72_5"/><span class="lab"><font color="Green" title="E56">A54</font></span>: 
<font color="Maroon" title="c15">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c17">p</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:72_5"/><span class="lab"><font color="Green" title="E57">A55</font></span>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E46">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:72_5"/><span class="lab"><font color="Green" title="E58">A56</font></span>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">and </span><br/><a NAME="E7:72_5"/>
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_5"><span class="lab"><font color="Green" title="E57">A55</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E9:72_5"/><span class="lab"><font color="Green" title="E59">A57</font></span>: 
<font color="Maroon" title="c18">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h</font>
 <span class="kw">and </span><br/><a NAME="E10:72_5"/><span class="lab"><font color="Green" title="E60">A58</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">and </span><br/><a NAME="E11:72_5"/><span class="lab"><font color="Green" title="E61">A59</font></span>: 
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>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font> holds <br/><font color="Maroon" title="c19">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72_5"><span class="lab"><font color="Green" title="E58">A56</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">h</font> = <font color="Maroon" title="c19">h</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72_5"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="txt" href="pua2mss1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E13:72_5"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c17">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T10" title="FUNCT_6:th.10">FUNCT_6:10</a></span>;<br/>
<a NAME="E14:72_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A60</font></span>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">p</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c17">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72_5"><span class="lab"><font color="Green" title="E58">A56</font></span></a>, <a class="txt" href="#E9:72_5"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E15:72_5"/><span class="lab"><font color="Green" title="E63">A61</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c17">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>;<br/>
<a NAME="E16:72_5"/><span class="lab"><font color="Green" title="E64">A62</font></span>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E17:72_5"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c2">P</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="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></span>;<br/>
<a NAME="E18:72_5"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c20">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:72_5"><span class="lab"><font color="Green" title="E62">A60</font></span></a>, <a class="txt" href="#E15:72_5"><span class="lab"><font color="Green" title="E63">A61</font></span></a>, <a class="txt" href="#E16:72_5"><span class="lab"><font color="Green" title="E64">A62</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E19:72_5"/><span class="kw">then </span>
<font color="Maroon" title="c20">h</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="pua2mss1.html#K10" title="PUA2MSS1:func.10">rng</a> <font color="Maroon" title="c8">R</font><span class="kw"> such that </span><br/><a NAME="E21:72_5"/><span class="lab"><font color="Green" title="E65">A63</font></span>: 
<font color="Maroon" title="c20">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c21">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#T7" target="_self" title="PUA2MSS1:th.7">Th7</a></span>;<br/>
<a NAME="E22:72_5"/><span class="lab"><font color="Green" title="E66">A64</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:72_5"><span class="lab"><font color="Green" title="E65">A63</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/>
<a NAME="E23:72_5"/><span class="lab"><font color="Green" title="E67">A65</font></span>: 
 <a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font> <a href="pua2mss1.html#R2" title="PUA2MSS1:pred.2">is_exactly_partitable_wrt</a> <font color="Maroon" title="c2">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pua2mss1.html#D11" target="_self" title="PUA2MSS1:def.11">Def11</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">r</font> implies <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:72_5_1"/><span class="lab"><font color="Green" title="E68">A66</font></span>: 
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font></span><br/><a NAME="E2:72_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A67</font></span>: 
<font color="Maroon" title="c20">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72_5"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="txt" href="#E11:72_5"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="txt" href="#E22:72_5"><span class="lab"><font color="Green" title="E66">A64</font></span></a></span>;<br/><a NAME="E3:72_5_1"/><span class="lab"><font color="Green" title="E70">A68</font></span>: 
<font color="Maroon" title="c20">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:72_5"><span class="lab"><font color="Green" title="E65">A63</font></span></a>, <a class="txt" href="#E1:72_5_1"><span class="lab"><font color="Green" title="E68">A66</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><a NAME="E4:72_5_1"/><span class="lab"><font color="Green" title="E71">A69</font></span>: 
<font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c21">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_5_1"><span class="lab"><font color="Green" title="E68">A66</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:72_5_1"/><span class="lab"><font color="Green" title="E72">A70</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72_5"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="txt" href="#E22:72_5"><span class="lab"><font color="Green" title="E66">A64</font></span></a>, <a class="txt" href="#E1:72_5_1"><span class="lab"><font color="Green" title="E68">A66</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E6:72_5_1"/>
 ex <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:72"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E4:72_5_1"><span class="lab"><font color="Green" title="E71">A69</font></span></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E7:72_5_1"/>
<font color="Maroon" title="c21">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72_5_1"><span class="lab"><font color="Green" title="E69">A67</font></span></a>, <a class="txt" href="#E3:72_5_1"><span class="lab"><font color="Green" title="E70">A68</font></span></a>, <a class="txt" href="#E5:72_5_1"><span class="lab"><font color="Green" title="E72">A70</font></span></a>, <a class="ref" href="pua2mss1.html#T3" target="_self" title="PUA2MSS1:th.3">Th3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E25:72_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A71</font></span>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c21">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72_5"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="txt" href="#E22:72_5"><span class="lab"><font color="Green" title="E66">A64</font></span></a>, <a class="ref" href="card_3.html#T38" title="CARD_3:th.38">CARD_3:38</a></span>;<br/>
<a NAME="E26:72_5"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c17">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pua2mss1.html#K2" title="PUA2MSS1:func.2">Den</a> <font color="Maroon" title="c16">o</font>,<font color="Maroon" title="c1">A</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_5"><span class="lab"><font color="Green" title="E57">A55</font></span></a>, <a class="txt" href="#E23:72_5"><span class="lab"><font color="Green" title="E67">A65</font></span></a>, <a class="ref" href="pua2mss1.html#D10" target="_self" title="PUA2MSS1:def.10">Def10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">s2</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E28:72_5"/><span class="lab"><font color="Green" title="E69">A72</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c22">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">r</font>
 <span class="kw">and </span><br/><a NAME="E29:72_5"/><span class="lab"><font color="Green" title="E70">A73</font></span>: 
<font color="Maroon" title="c22">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E25:72_5"><span class="lab"><font color="Green" title="E68">A71</font></span></a>, <a class="ref" href="pua2mss1.html#T33" target="_self" title="PUA2MSS1:th.33">Th33</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">p9</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c24">o9</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c4">S2</font><span class="kw"> such that </span><br/><a NAME="E31:72_5"/><span class="lab"><font color="Green" title="E71">A74</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c22">s2</font></span>)</span>,<font color="Maroon" title="c23">p9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E32:72_5"/><span class="lab"><font color="Green" title="E72">A75</font></span>: 
<font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">o9</font>
 <span class="kw">and </span><br/><a NAME="E33:72_5"/><span class="lab"><font color="Green" title="E73">A76</font></span>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c24">o9</font>,<font color="Maroon" title="c7">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c23">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:72"><span class="lab"><font color="Green" title="E54">A22</font></span></a></span>;<br/>
<a NAME="E34:72_5"/>
 <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">S2</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">S2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span> <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c22">s2</font>,<font color="Maroon" title="c7">G</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c7">G</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c4">S2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c22">s2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c21">r</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:72_5"><span class="lab"><font color="Green" title="E69">A72</font></span></a>, <a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></span>
;<br/>
<a NAME="E36:72_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A77</font></span>: 
<font color="Maroon" title="c23">p9</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">em</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:72"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E32:72_5"><span class="lab"><font color="Green" title="E72">A75</font></span></a>, <a class="txt" href="#E33:72_5"><span class="lab"><font color="Green" title="E73">A76</font></span></a>, <a class="ref" href="pua2mss1.html#T5" target="_self" title="PUA2MSS1:th.5">Th5</a></span>;<br/>
<a NAME="E37:72_5"/><span class="lab"><font color="Green" title="E75">A78</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c11">em</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c21">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:72"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E25:72_5"><span class="lab"><font color="Green" title="E68">A71</font></span></a>, <a class="ref" href="pua2mss1.html#T5" target="_self" title="PUA2MSS1:th.5">Th5</a></span>;<br/>
<a NAME="E38:72_5"/>
<font color="Maroon" title="c6">Q</font> <a href="pua2mss1.html#K5" title="PUA2MSS1:func.5">-index_of</a> <font color="Maroon" title="c22">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E29:72_5"><span class="lab"><font color="Green" title="E70">A73</font></span></a>, <a class="ref" href="pua2mss1.html#D4" target="_self" title="PUA2MSS1:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E39:72_5"/>
<font color="Maroon" title="c15">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:72"><span class="lab"><font color="Green" title="E54">A22</font></span></a>, <a class="txt" href="#E3:72_5"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="txt" href="#E31:72_5"><span class="lab"><font color="Green" title="E71">A74</font></span></a>, <a class="txt" href="#E36:72_5"><span class="lab"><font color="Green" title="E74">A77</font></span></a>, <a class="txt" href="#E37:72_5"><span class="lab"><font color="Green" title="E75">A78</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


</div>
