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<b>let </b><font color="Maroon" title="c1">I1</font>, <font color="Maroon" title="c2">I2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">I2</font>;<br/><b>let </b><font color="Maroon" title="c5">o</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I2</font>;<br/>







<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">o</font>;<br/><b>let </b><font color="Maroon" title="c7">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c2">I2</font>;<br/>



<b>assume </b><a NAME="E2:29"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c1">I1</font> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c5">o</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c6">m</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">Xm</font> = <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="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">o'</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o'</font>] </span>}</span>  as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>from </b><i><a class="ref" href="altcat_2.html#S1" title="ALTCAT_2:sch.1">ALTCAT_2:sch 1</a>();<br/></i>
<a NAME="E4:29"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c8">Xm</font> <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="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">o'</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o'</font>] </span>}</span> 
 
;<br/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">Xm</font> = 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">o'</font> where <font color="Olive" title="b1">o'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o'</font>] </span>}</span> 
<b>from </b><i><a class="ref" href="altcat_2.html#S2" title="ALTCAT_2:sch.2">ALTCAT_2:sch 2</a>(<a class="txt" href="functor0.html#E4:29"><i><font color="Green" title="E24">A3</font></i></a>)</i>
<br/>.= 
<font color="Maroon" title="c1">I1</font>
<b>by </b><i><a class="ref" href="domain_1.html#T48" title="DOMAIN_1:th.48">DOMAIN_1:48</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">Xm</font> = <font color="Maroon" title="c8">Xm</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I1</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">m</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c6">m</font>;<br/>
<a NAME="E7:29"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c9">Xm</font> <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="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">o'</font>)</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I1</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o'</font>] </span>}</span> 
 
;<br/>
<a NAME="E8:29"/>
<font color="Maroon" title="c9">Xm</font> is    <a href="functor0.html#M1" title="FUNCTOR0:mode.1">MSUnTrans</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2_1"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c11">o'</font> = <font color="Maroon" title="c10">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I1</font> ;<br/><a NAME="E3:29_2_1"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">o'</font>]
 ;<br/><a NAME="E4:29_2_1"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="txt" href="functor0.html#E2:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="functor0.html#E1:29_2_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E5:29_2_1"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o</font>
 <b>by </b><i><a class="txt" href="functor0.html#E2:29"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="functor0.html#E1:29_2_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E6:29_2_1"/><b>then </b>
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="functor0.html#E1:29"><i><font color="Green" title="E22">A1</font></i></a></i>;<br/><a NAME="E7:29_2_1"/><b>then </b><i><font color="Green" title="E29">A8</font></i>: 
<font color="Maroon" title="c6">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="functor0.html#E4:29_2_1"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E8:29_2_1"/>
<font color="Maroon" title="c9">Xm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">o'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">o'</font>)
 <b>from </b><i><a class="ref" href="altcat_2.html#S3" title="ALTCAT_2:sch.3">ALTCAT_2:sch 3</a>(<a class="txt" href="functor0.html#E7:29"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="functor0.html#E3:29_2_1"><i><font color="Green" title="E27">A6</font></i></a>);<br/></i><b>hence </b><a NAME="E9:29_2_1"/>
<font color="Maroon" title="c9">Xm</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="functor0.html#E7:29_2_1"><i><font color="Green" title="E29">A8</font></i></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:29_2"/>
<font color="Maroon" title="c9">Xm</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> <font color="Maroon" title="c7">f</font>
 <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor0.html#D5" target="_self" title="FUNCTOR0:def.5">FUNCTOR0:def 5</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:29"/>
<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>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o'</font></span>)</span> <a href="pboole.html#K9" title="PBOOLE:func.9">--&gt;</a> <font color="Maroon" title="c6">m</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">o'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I1</font> : verum </span>}</span>  is    <a href="functor0.html#M1" title="FUNCTOR0:mode.1">MSUnTrans</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">B</font>
 ;<br/>


</div>
