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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>   <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>   <a href="altcat_2.html#V1" title="ALTCAT_2:attr.1">reflexive</a>   <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</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> being   <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  holds <br/> for <font color="Olive" title="b2">G</font> being   <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>  st  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Olive" title="b2">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Olive" title="b2">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  holds <br/><font color="Olive" title="b1">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">I1</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">I2</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</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">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  implies  for <font color="Olive" title="b1">G</font> being   <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>  st  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Olive" title="b1">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Olive" title="b1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  holds <br/><font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</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">G</font> being   <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>  st  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Olive" title="b1">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Olive" title="b1">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  holds <br/><font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">k</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">A</font>, the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font><span class="kw"> such that </span><br/><a NAME="E3:22"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E4:22"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="functor0.html#D38" title="FUNCTOR0:def.38">FUNCTOR0:def 38</a></span>;<br/>
<a NAME="E5:22"/>
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V19" title="FUNCTOR0:attr.19">injective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a></span>;<br/>
<a NAME="E6:22"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V4" title="FUNCTOR0:attr.4">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D33" title="FUNCTOR0:def.33">FUNCTOR0:def 33</a></span>;<br/>
<a NAME="E7:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A4</font></span>: 
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</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="ref" href="functor0.html#D6" title="FUNCTOR0:def.6">FUNCTOR0:def 6</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">OM</font> =  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font>;<br/>
<a NAME="E8:22"/>
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V20" title="FUNCTOR0:attr.20">surjective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a></span>;<br/>
<a NAME="E9:22"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D34" title="FUNCTOR0:def.34">FUNCTOR0:def 34</a></span>;<br/>
<a NAME="E10:22"/><span class="kw">then </span>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D7" title="FUNCTOR0:def.7">FUNCTOR0:def 7</a></span>;<br/>
<a NAME="E11:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E12:22"/>
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V19" title="FUNCTOR0:attr.19">injective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a></span>;<br/>
<a NAME="E13:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D33" title="FUNCTOR0:def.33">FUNCTOR0:def 33</a></span>;<br/>
<a NAME="E14:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">F</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D30" title="FUNCTOR0:def.30">FUNCTOR0:def 30</a></span>;<br/>
<a NAME="E15:22"/>
<font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V20" title="FUNCTOR0:attr.20">surjective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a></span>;<br/>
<a NAME="E16:22"/><span class="kw">then </span>
( <font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V18" title="FUNCTOR0:attr.18">full</a>  &amp; <font color="Maroon" title="c5">F</font> is  <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D34" title="FUNCTOR0:def.34">FUNCTOR0:def 34</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">A</font>, the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font><span class="kw"> such that </span><br/><a NAME="E18:22"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E19:22"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D32" title="FUNCTOR0:def.32">FUNCTOR0:def 32</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">G</font> be   <a href="functor0.html#V8" title="FUNCTOR0:attr.8">feasible</a>   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c9">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</a>  implies <font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font> )</span><br/>

<span class="kw">assume </span><a NAME="E20:22"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
 <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c9">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K15" title="FUNCTOR0:func.15">"</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="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font></span><br/>

<a NAME="E21:22"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</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="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E20:22"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="functor0.html#D38" title="FUNCTOR0:def.38">FUNCTOR0:def 38</a></span>;<br/>
<a NAME="E22:22"/>
 <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c9">G</font> #) is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E20:22"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="functor0.html#T35" title="FUNCTOR0:th.35">FUNCTOR0:35</a></span>;<br/>
<a NAME="E23:22"/><span class="kw">then </span>
 <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c9">G</font> #) is  <a href="functor0.html#V20" title="FUNCTOR0:attr.20">surjective</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D35" title="FUNCTOR0:def.35">FUNCTOR0:def 35</a></span>;<br/>
<a NAME="E24:22"/><span class="kw">then </span>
(  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c9">G</font> #) is  <a href="functor0.html#V18" title="FUNCTOR0:attr.18">full</a>  &amp;  <a href="functor0.html#G2" title="FUNCTOR0:aggr.2">FunctorStr</a>(#  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>, the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <font color="Maroon" title="c9">G</font> #) is  <a href="functor0.html#V5" title="FUNCTOR0:attr.5">onto</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D34" title="FUNCTOR0:def.34">FUNCTOR0:def 34</a></span>;<br/>
<a NAME="E25:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A12</font></span>: 
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D7" title="FUNCTOR0:def.7">FUNCTOR0:def 7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">OMG</font> =  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>;<br/>
<a NAME="E26:22"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">OM</font> =  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> as   <a href="functor0.html#NM1" title="FUNCTOR0:NM.1">bifunction</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font> ;<br/>
<a NAME="E28:22"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span></span>)</span>
 
<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="E29:22"/><span class="lab"><font color="Green" title="E25">A15</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span></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> <span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">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="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:22_1"/><span class="lab"><font color="Green" title="E26">A16</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span><br/>

<a NAME="E2:22_1"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<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="E3:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>;<br/>
<a NAME="E4:22_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:22"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E5:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A19</font></span>: 
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A20</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></span>
<br/>.= 
 the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:22"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
;<br/>
<span class="lab"><font color="Green" title="E31">A21</font></span>:  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">OM</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">OM</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:22"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_1.html#T39" title="FUNCT_1:th.39">FUNCT_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
;<br/>
<a NAME="E8:22_1"/>
( <font color="Maroon" title="c8">f</font> is  <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:22"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="functor0.html#D30" title="FUNCTOR0:def.30">FUNCTOR0:def 30</a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E9:22_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A22</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span> 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="#E5:22_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</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> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E2:22_1"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="funct_1.html#T13" title="FUNCT_1:th.13">FUNCT_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:22"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E19:22"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E5:22_1"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="ref" href="msualg_3.html#D4" title="MSUALG_3:def.4">MSUALG_3:def 4</a></span>
;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:22"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E28:22"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E26">A16</font></span></a>, <a class="txt" href="#E3:22_1"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="pboole.html#D19" title="PBOOLE:def.19">PBOOLE:def 19</a></span>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:22_1"><span class="lab"><font color="Green" title="E32">A22</font></span></a>, <a class="txt" href="#E6:22_1"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E7:22_1"><span class="lab"><font color="Green" title="E31">A21</font></span></a>, <a class="ref" href="funct_1.html#T39" title="FUNCT_1:th.39">FUNCT_1:39</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22_1"><span class="lab"><font color="Green" title="E26">A16</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="E12:22_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">i</font>
 ; <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="E30:22"/>
 the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#K8" title="PBOOLE:func.8">**</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="msualg_3.html#K4" title="MSUALG_3:func.4">""</a></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:22"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E4:22"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E18:22"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E21:22"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="ref" href="functor0.html#D36" title="FUNCTOR0:def.36">FUNCTOR0:def 36</a></span>;<br/>
<a NAME="E31:22"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A23</font></span>: 
 the <a href="functor0.html#U2" title="FUNCTOR0:sel.2">MorphMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:22"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</a></span>;<br/>
<a NAME="E32:22"/>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c9">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D36" title="FUNCTOR0:def.36">FUNCTOR0:def 36</a></span>;<br/>
<a NAME="E33:22"/><span class="kw">then </span>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"> the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <font color="Maroon" title="c5">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E20:22"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="functor0.html#D38" title="FUNCTOR0:def.38">FUNCTOR0:def 38</a></span>;<br/>
<a NAME="E34:22"/><span class="kw">then </span>
 the <a href="functor0.html#U1" title="FUNCTOR0:sel.1">ObjectMap</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font></span>)</span> <a href="binop_1.html#R8" title="BINOP_1:pred.8">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:22"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="txt" href="#E11:22"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="funct_1.html#T39" title="FUNCT_1:th.39">FUNCT_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E35:22"/>
<font color="Maroon" title="c5">F</font> <a href="functor0.html#K13" title="FUNCTOR0:func.13">*</a> <font color="Maroon" title="c9">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K12" title="FUNCTOR0:func.12">id</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:22"><span class="lab"><font color="Green" title="E26">A23</font></span></a>, <a class="ref" href="functor0.html#D29" title="FUNCTOR0:def.29">FUNCTOR0:def 29</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>
