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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  <a href="instalg1.html#V1" title="INSTALG1:attr.1">feasible</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ;<br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E14">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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">S</font> &amp; the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> &amp; the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font>;<br/>
<b>set </b><font color="Maroon" title="c4">g</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font>;<br/>
<b>thus </b><a NAME="E2:19"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span>)</span> <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="c2">T</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font> )
 <b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pua2mss1.html#D13" title="PUA2MSS1:def.13">PUA2MSS1:def 13</a>,<a class="ref" href="instalg1.html#D2" target="_self" title="INSTALG1:def.2">INSTALG1:def 2</a><br/>

<b>thus </b><a NAME="E3:19"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span>)</span> <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">S</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>

<a NAME="E4:19"/><i><font color="Green" title="E15">A2</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:19"/><i><font color="Green" title="E16">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="instalg1.html#T8" target="_self" title="INSTALG1:th.8">Th8</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green" title="E17">A4</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">T</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="c2">T</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</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">S</font> )
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E7:19"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E8:19"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="instalg1.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="instalg1.html#E4:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>;<br/>

<b>thus </b><span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">T</font> = 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="instalg1.html#E6:19"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font>
<b>by </b><i><a class="txt" href="instalg1.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="instalg1.html#E5:19"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="grfunc_1.html#T64" title="GRFUNC_1:th.64">GRFUNC_1:64</a></i>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></i>
;<br/>

<b>let </b><font color="Maroon" title="c5">o</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>



<b>assume </b><a NAME="E10:19"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c5">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c2">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">q</font> = <font color="Maroon" title="c6">p</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="c2">T</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E12:19"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">q</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="c2">T</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">p</font> = 
<font color="Maroon" title="c6">p</font>
<b>by </b><i><a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>
<br/>.= 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font>
<b>by </b><i><a class="txt" href="instalg1.html#E1:19"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="instalg1.html#E4:19"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="instalg1.html#E10:19"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">OperSymbols</a> of <font color="Maroon" title="c2">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font></span>)</span>
<b>by </b><i><a class="txt" href="instalg1.html#E10:19"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="funct_1.html#T34" title="FUNCT_1:th.34">FUNCT_1:34</a></i>
;<br/><br/>


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