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<b>let </b><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V2</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c5">F</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>;<br/><b>let </b><font color="Maroon" title="c6">G</font> be    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c6">G</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Maroon" title="c6">G</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <b>by </b><i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></i>;<br/><a NAME="E4:22_1"/><i><font color="Green" title="E18">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></i>;<br/><a NAME="E5:22_1"/><i><font color="Green" title="E19">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></i>;<br/><a NAME="E6:22_1"/><i><font color="Green" title="E20">A6</font></i>: 
( <font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> &amp; <font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="autalg_1.html#E2:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="autalg_1.html#T16" target="_self" title="AUTALG_1:th.16">Th16</a>, <a class="ref" href="autalg_1.html#T17" target="_self" title="AUTALG_1:th.17">Th17</a></i>;<br/><a NAME="E7:22_1"/><i><font color="Green" title="E21">A7</font></i>: 
<font color="Maroon" title="c8">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></i>;<br/><a NAME="E8:22_1"/><i><font color="Green" title="E22">A8</font></i>: 
<font color="Maroon" title="c9">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#T1" title="MSUALG_3:th.1">MSUALG_3:1</a></i>;<br/><a NAME="E9:22_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>;<br/><a NAME="E10:22_1"/><b>then </b><i><font color="Green" title="E23">A9</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="txt" href="autalg_1.html#E6:22_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></i>;<br/><a NAME="E11:22_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">ff</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <b>by </b><i><a class="txt" href="autalg_1.html#E4:22_1"><i><font color="Green" title="E18">A4</font></i></a>, <a class="txt" href="autalg_1.html#E7:22_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/><a NAME="E13:22_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">gg</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <b>by </b><i><a class="txt" href="autalg_1.html#E5:22_1"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="autalg_1.html#E8:22_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<font color="Maroon" title="c10">ff</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">gg</font>
<b>by </b><i><a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</a></i>
<br/>.= 
<font color="Maroon" title="c10">ff</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E2:22"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="autalg_1.html#E1:22"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="autalg_1.html#E1:22_1"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></i>
;<br/><b>hence </b><a NAME="E16:22_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="autalg_1.html#E7:22_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="txt" href="autalg_1.html#E8:22_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="txt" href="autalg_1.html#E10:22_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></i>;<br/>


</div>
