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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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</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="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> <font color="Olive" title="b2">B</font> holds <br/>for <font color="Olive" title="b3">H</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font><br/> for <font color="Olive" title="b4">H1</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Olive" title="b2">B</font>,<font color="Olive" title="b1">A</font>  st <font color="Olive" title="b3">H</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp; <font color="Olive" title="b3">H</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  &amp; <font color="Olive" title="b4">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b3">H</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  holds <br/>( <font color="Olive" title="b3">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Olive" title="b4">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b4">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Olive" title="b3">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Olive" title="b1">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">B</font> implies for <font color="Olive" title="b1">H</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><br/> for <font color="Olive" title="b2">H1</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">H</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp; <font color="Olive" title="b1">H</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  &amp; <font color="Olive" title="b2">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  holds <br/>( <font color="Olive" title="b1">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Olive" title="b2">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b2">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Olive" title="b1">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">A</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:25"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c2">A</font> <a href="pzfmisc1.html#R1" title="PZFMISC1:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">B</font>
 ; <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">H</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font><br/> for <font color="Olive" title="b2">H1</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">A</font>  st <font color="Olive" title="b1">H</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp; <font color="Olive" title="b1">H</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  &amp; <font color="Olive" title="b2">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">H</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  holds <br/>( <font color="Olive" title="b1">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Olive" title="b2">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b2">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Olive" title="b1">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">H</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font>; <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">H1</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">A</font>  st <font color="Maroon" title="c4">H</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp; <font color="Maroon" title="c4">H</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  &amp; <font color="Olive" title="b1">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  holds <br/>( <font color="Maroon" title="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Olive" title="b1">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b1">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">H1</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c2">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="c4">H</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp; <font color="Maroon" title="c4">H</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  &amp; <font color="Maroon" title="c5">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a>  implies ( <font color="Maroon" title="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c5">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">A</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:25"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
( <font color="Maroon" title="c4">H</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp; <font color="Maroon" title="c4">H</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>  )
 <span class="kw">and </span><br/><a NAME="E3:25"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c5">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c4">H</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</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="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c5">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">F1</font> = <font color="Maroon" title="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c5">H1</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<div><span class="lab"><font color="Green" title="E15">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c5">H1</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>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><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></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:25_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<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>
 ; <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="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c5">H1</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>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><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></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c4">H</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 <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><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></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">h1</font> = <font color="Maroon" title="c5">H1</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 <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><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></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/><a NAME="E4:25_1"/>
<font color="Maroon" title="c7">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="c4">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><a NAME="E5:25_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<font color="Maroon" title="c8">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a></span>;<br/><a NAME="E6:25_1"/>
<font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></span>;<br/><a NAME="E7:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">h</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></span>;<br/><a NAME="E8:25_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E9:25_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c5">H1</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>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><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></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:25_2"/>
<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>
 ; <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="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><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></span>)</span></span><br/><a NAME="E2:25_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">F1</font> <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>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E15">A4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_2"/>
<font color="Maroon" title="c6">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><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></span>)</span>,<span class="p1">(<span class="default"><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></span>)</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="E7:25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c6">F1</font> is    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<a NAME="E9:25"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:25_3"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c8">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">h</font> = <font color="Maroon" title="c4">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">h1</font> = <font color="Maroon" title="c5">H1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E4:25_3"/>
<font color="Maroon" title="c8">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="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E5:25_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c9">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a></span>;<br/>
<a NAME="E6:25_3"/>
( <font color="Maroon" title="c3">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="pzfmisc1.html#D3" title="PZFMISC1:def.3">PZFMISC1:def 3</a></span>;<br/>
<a NAME="E7:25_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</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="E8:25_3"/>
<font color="Maroon" title="c10">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:25"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E3:25"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="msualg_3.html#D5" title="MSUALG_3:def.5">MSUALG_3:def 5</a></span>;<br/>
<a NAME="E9:25_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">h1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25_3"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E7:25_3"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:25_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:25_3"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="msualg_3.html#T2" title="MSUALG_3:th.2">MSUALG_3:2</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:25_4"/>
<font color="Maroon" title="c8">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="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span></span><br/><a NAME="E2:25_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E17">A8</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:25_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c7">F</font> as    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<a NAME="E12:25"/>
<font color="Maroon" title="c8">F</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:25"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="msualg_3.html#D1" title="MSUALG_3:def.1">MSUALG_3:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:25"/>
( <font color="Maroon" title="c4">H</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c5">H1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">H1</font> <a href="altcat_2.html#K2" title="ALTCAT_2:func.2">**</a> <font color="Maroon" title="c4">H</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a> <font color="Maroon" title="c2">A</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:25"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E7:25"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="msualg_3.html#D1" title="MSUALG_3:def.1">MSUALG_3:def 1</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/>


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