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<span class="kw">let </span><font color="Maroon" title="c1">I</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">I</font><br/>  for <font color="Olive" title="b3">A</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b3">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Olive" title="b2">G</font> holds <br/><font color="Olive" title="b1">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Olive" title="b3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Olive" title="b3">A</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">A</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c3">G</font> holds <br/><font color="Maroon" title="c2">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Olive" title="b1">A</font></span><br/>



<a NAME="E1:8"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">FG</font> = <font color="Maroon" title="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T15" title="MSSUBFAM:th.15">MSSUBFAM:15</a></span>;<br/>
<a NAME="E3:8"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E4:8"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">FG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">A</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c3">G</font> implies <font color="Maroon" title="c2">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c5">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:8"/><span class="lab"><font color="Green" title="E7">A4</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="pboole.html#R1" title="PBOOLE:pred.1">in</a>  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c3">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c5">A</font></span><br/>

<div><span class="lab"><font color="Green" title="E8">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">i</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c3">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">fg</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><span class="kw">assume </span><a NAME="E4:8_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span><br/><a NAME="E5:8_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mssubfam.html#T14" title="MSSUBFAM:th.14">MSSUBFAM:14</a></span>;<br/><a NAME="E6:8_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A7</font></span>: 
<font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="pboole.html#D1" title="PBOOLE:def.1">PBOOLE:def 1</a></span>;<br/><a NAME="E7:8_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> = 
<font color="Maroon" title="c7">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">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:8"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">g</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="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<font color="Maroon" title="c9">fg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E4:8_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E7:8"/>
<font color="Maroon" title="c4">FG</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font> is   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E8:8"/>
<font color="Maroon" title="c2">F</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <font color="Maroon" title="c3">G</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c5">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
