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<b>consider </b><font color="Maroon" title="c1">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:8_1"/><i><font color="Green" title="E4">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>consider </b><font color="Maroon" title="c2">a0</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E4:8_1"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">a0</font>]
 <b>by </b><i><a class="txt" href="yellow18.html#E1:8"><i><font color="Green" title="E1">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">X</font> = <font color="Maroon" title="c1">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="yellow18.html#E4:8_1"><i><font color="Green" title="E5">A5</font></i></a></i>;<br/>
<a NAME="E6:8_1"/><i><font color="Green" title="E6">A6</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:8_1_1"/>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> $1,$2;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> $1,$2,$3</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> $5,$4;<br/>
<div><i><font color="Green" title="E7">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font>;<br/><b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:8_1_2"/><i><font color="Green" title="E8">A8</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">f</font>] )
 <b>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green" title="E9">A9</font></i>: 
( <font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">g</font>] )
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c11">c'</font> = <font color="Maroon" title="c6">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><a NAME="E4:8_1_2"/><i><font color="Green" title="E10">A10</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c12">f'</font> = <font color="Maroon" title="c7">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:8_1_2"><i><font color="Green" title="E8">A8</font></i></a></i>;<br/><a NAME="E6:8_1_2"/><i><font color="Green" title="E11">A11</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c13">g'</font> = <font color="Maroon" title="c8">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font> <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1_2"><i><font color="Green" title="E9">A9</font></i></a></i>;<br/><a NAME="E8:8_1_2"/><i><font color="Green" title="E12">A12</font></i>: 
( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f'</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c11">c'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:8_1_2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="txt" href="yellow18.html#E2:8_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="yellow18.html#E4:8_1_2"><i><font color="Green" title="E10">A10</font></i></a>, <a class="txt" href="yellow18.html#E6:8_1_2"><i><font color="Green" title="E11">A11</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>;<br/><b>hence </b><a NAME="E9:8_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>)
 ;<br/><a NAME="E10:8_1_2"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c9">a'</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c10">b'</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c11">c'</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E11:8_1_2"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>)]
 <b>by </b><i><a class="txt" href="yellow18.html#E2:8"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="yellow18.html#E1:8_1_2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="txt" href="yellow18.html#E2:8_1_2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="txt" href="yellow18.html#E8:8_1_2"><i><font color="Green" title="E12">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E8">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font>;<br/><b>let </b><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font>, <font color="Maroon" title="c10">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:8_1_3"/><i><font color="Green" title="E9">A14</font></i>: 
( <font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">f</font>] )
 <b>and </b><br/><a NAME="E2:8_1_3"/><i><font color="Green" title="E10">A15</font></i>: 
( <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c9">g</font>] )
 <b>and </b><br/><a NAME="E3:8_1_3"/><i><font color="Green" title="E11">A16</font></i>: 
( <font color="Maroon" title="c10">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c10">h</font>] )
 ;<br/><b>reconsider </b><font color="Maroon" title="c11">a'</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c12">b'</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c13">c'</font> = <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c14">d'</font> = <font color="Maroon" title="c7">d</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><a NAME="E5:8_1_3"/><i><font color="Green" title="E12">A17</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">a'</font>,<font color="Maroon" title="c12">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c15">f'</font> = <font color="Maroon" title="c8">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">a'</font>,<font color="Maroon" title="c12">b'</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:8_1_3"><i><font color="Green" title="E9">A14</font></i></a></i>;<br/><a NAME="E7:8_1_3"/><i><font color="Green" title="E13">A18</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c12">b'</font>,<font color="Maroon" title="c13">c'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c16">g'</font> = <font color="Maroon" title="c9">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c12">b'</font>,<font color="Maroon" title="c13">c'</font> <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1_3"><i><font color="Green" title="E10">A15</font></i></a></i>;<br/><a NAME="E9:8_1_3"/><i><font color="Green" title="E14">A19</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">d'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c17">h'</font> = <font color="Maroon" title="c10">h</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">d'</font> <b>by </b><i><a class="txt" href="yellow18.html#E3:8_1_3"><i><font color="Green" title="E11">A16</font></i></a></i>;<br/><a NAME="E11:8_1_3"/><i><font color="Green" title="E15">A20</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c11">a'</font>,<font color="Maroon" title="c13">c'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c12">b'</font>,<font color="Maroon" title="c14">d'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:8_1_3"><i><font color="Green" title="E9">A14</font></i></a>, <a class="txt" href="yellow18.html#E2:8_1_3"><i><font color="Green" title="E10">A15</font></i></a>, <a class="txt" href="yellow18.html#E3:8_1_3"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="yellow18.html#E5:8_1_3"><i><font color="Green" title="E12">A17</font></i></a>, <a class="txt" href="yellow18.html#E7:8_1_3"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="yellow18.html#E9:8_1_3"><i><font color="Green" title="E14">A19</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c9">g</font>),<font color="Maroon" title="c10">h</font>) = 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">a'</font>,<font color="Maroon" title="c13">c'</font>,<font color="Maroon" title="c14">d'</font>,<font color="Maroon" title="c16">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">f'</font>,<font color="Maroon" title="c10">h</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E1:8_1_3"><i><font color="Green" title="E9">A14</font></i></a>, <a class="txt" href="yellow18.html#E2:8_1_3"><i><font color="Green" title="E10">A15</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c17">h'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">f'</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E3:8_1_3"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="yellow18.html#E11:8_1_3"><i><font color="Green" title="E15">A20</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">g'</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">f'</font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:8_1_3"><i><font color="Green" title="E9">A14</font></i></a>, <a class="txt" href="yellow18.html#E2:8_1_3"><i><font color="Green" title="E10">A15</font></i></a>, <a class="txt" href="yellow18.html#E3:8_1_3"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">f</font>,<font color="Maroon" title="c17">h'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">g'</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E1:8_1_3"><i><font color="Green" title="E9">A14</font></i></a>, <a class="txt" href="yellow18.html#E11:8_1_3"><i><font color="Green" title="E15">A20</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c8">f</font>,<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">d</font>,<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c10">h</font>))
<b>by </b><i><a class="txt" href="yellow18.html#E2:8_1_3"><i><font color="Green" title="E10">A15</font></i></a>, <a class="txt" href="yellow18.html#E3:8_1_3"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E9">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font>;<br/><b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">f</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">b</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font>;<br/><a NAME="E3:8_1_4"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c5">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">b</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:8_1_4"/>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">f</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E3:8"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c8">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font>;<br/><b>let </b><font color="Maroon" title="c9">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E5:8_1_4"/><i><font color="Green" title="E10">A22</font></i>: 
( <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">c</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">g</font>] )
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">d</font> = <font color="Maroon" title="c8">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c9">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c10">d</font> <b>by </b><i><a class="txt" href="yellow18.html#E5:8_1_4"><i><font color="Green" title="E10">A22</font></i></a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">g</font>) = 
<font color="Maroon" title="c11">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">b</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E5:8_1_4"><i><font color="Green" title="E10">A22</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:8_1_4"><i><font color="Green" title="E10">A22</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E10">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font>;<br/><b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">f</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">b</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font>;<br/><a NAME="E3:8_1_5"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c5">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">b</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E4:8_1_5"/>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">f</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E3:8"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c8">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">X</font>;<br/><b>let </b><font color="Maroon" title="c9">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E5:8_1_5"/><i><font color="Green" title="E11">A24</font></i>: 
( <font color="Maroon" title="c9">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c4">a</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">g</font>] )
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">d</font> = <font color="Maroon" title="c8">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c9">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">d</font>,<font color="Maroon" title="c5">b</font> <b>by </b><i><a class="txt" href="yellow18.html#E5:8_1_5"><i><font color="Green" title="E11">A24</font></i></a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c7">f</font>) = 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c11">g'</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:8_1_5"><i><font color="Green" title="E11">A24</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:8_1_5"><i><font color="Green" title="E11">A24</font></i></a>, <a class="ref" href="altcat_1.html#T24" title="ALTCAT_1:th.24">ALTCAT_1:24</a></i>
;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">C</font> being  <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a><b> such that </b><br/><a NAME="E12:8_1"/><i><font color="Green" title="E11">A25</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font>
 <b>and </b><br/><a NAME="E13:8_1"/><i><font color="Green" title="E12">A26</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">C</font><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) )
 <b>and </b><br/><a NAME="E14:8_1"/><i><font color="Green" title="E13">A27</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">C</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b5">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> holds  <font color="Olive" title="b5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">g</font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S6" target="_self" title="YELLOW18:sch.6">YELLOW18:sch 6</a>(<a class="txt" href="yellow18.html#E7:8_1"><i><font color="Green" title="E7">A7</font></i></a>, <a class="txt" href="yellow18.html#E8:8_1"><i><font color="Green" title="E8">A13</font></i></a>, <a class="txt" href="yellow18.html#E9:8_1"><i><font color="Green" title="E9">A21</font></i></a>, <a class="txt" href="yellow18.html#E10:8_1"><i><font color="Green" title="E10">A23</font></i></a>);<br/></i>
<a NAME="E15:8_1"/>
<font color="Maroon" title="c4">C</font> is    <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:8_1_6"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="yellow18.html#E6:8_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="yellow18.html#E12:8_1"><i><font color="Green" title="E11">A25</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D11" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a><br/>

<b>thus </b><a NAME="E2:8_1_6"/>
<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="c4">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">F<sub>1</sub></font>(),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E6:8_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="yellow18.html#E12:8_1"><i><font color="Green" title="E11">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a><br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:8_1_6_1"/>
<font color="Maroon" title="c5">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="c4">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:8_1_6_1"/><i><font color="Green" title="E14">A28</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c7">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">C</font> <b>by </b><i><a class="txt" href="yellow18.html#E3:8_1_6_1"><i><font color="Green" title="E14">A28</font></i></a></i>;<br/><a NAME="E5:8_1_6_1"/><i><font color="Green" title="E15">A29</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E3:8_1_6_1"><i><font color="Green" title="E14">A28</font></i></a></i>;<br/><a NAME="E6:8_1_6_1"/><i><font color="Green" title="E16">A30</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E3:8_1_6_1"><i><font color="Green" title="E14">A28</font></i></a></i>;<br/><b>thus </b><a NAME="E7:8_1_6_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:8_1_6_1_1"/>
(  not <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> or <font color="Maroon" title="c10">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E13:8_1"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="yellow18.html#E5:8_1_6_1"><i><font color="Green" title="E15">A29</font></i></a>, <a class="txt" href="yellow18.html#E6:8_1_6_1"><i><font color="Green" title="E16">A30</font></i></a></i>;<br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E4:8_1_6"/>
<span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E6:8_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="yellow18.html#E12:8_1"><i><font color="Green" title="E11">A25</font></i></a>, <a class="ref" href="mcart_1.html#T77" title="MCART_1:th.77">MCART_1:77</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a><br/>

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E5:8_1_6"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:8_1_6"/><i><font color="Green" title="E14">A31</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">C</font> &amp; <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T72" title="MCART_1:th.72">MCART_1:72</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c8">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">C</font> <b>by </b><i><a class="txt" href="yellow18.html#E7:8_1_6"><i><font color="Green" title="E14">A31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">a'</font> = <font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c13">b'</font> = <font color="Maroon" title="c10">b</font>, <font color="Maroon" title="c14">c'</font> = <font color="Maroon" title="c11">c</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="yellow18.html#E12:8_1"><i><font color="Green" title="E11">A25</font></i></a></i>;<br/>
<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E10:8_1_6"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 ;<br/>

<a NAME="E11:8_1_6"/><b>then </b><i><font color="Green" title="E15">A32</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:8_1_6"><i><font color="Green" title="E14">A31</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">gf</font>, <font color="Maroon" title="c17">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:8_1_6"/><i><font color="Green" title="E16">A33</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">gf</font>,<font color="Maroon" title="c17">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E14:8_1_6"/><i><font color="Green" title="E17">A34</font></i>: 
<font color="Maroon" title="c16">gf</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"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E15:8_1_6"/><i><font color="Green" title="E18">A35</font></i>: 
<font color="Maroon" title="c17">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T6" title="RELSET_1:th.6">RELSET_1:6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">g</font>, <font color="Maroon" title="c19">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E17:8_1_6"/><i><font color="Green" title="E19">A36</font></i>: 
( <font color="Maroon" title="c18">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font> &amp; <font color="Maroon" title="c19">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">g</font>,<font color="Maroon" title="c19">f</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">gf</font> )
 <b>by </b><i><a class="txt" href="yellow18.html#E14:8_1_6"><i><font color="Green" title="E17">A34</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">f</font> = <font color="Maroon" title="c19">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> <b>by </b><i><a class="txt" href="yellow18.html#E17:8_1_6"><i><font color="Green" title="E19">A36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">g</font> = <font color="Maroon" title="c18">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font> <b>by </b><i><a class="txt" href="yellow18.html#E17:8_1_6"><i><font color="Green" title="E19">A36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">h</font> = <font color="Maroon" title="c17">h</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">c</font> <b>by </b><i><a class="txt" href="yellow18.html#E15:8_1_6"><i><font color="Green" title="E18">A35</font></i></a></i>;<br/>
<a NAME="E21:8_1_6"/><i><font color="Green" title="E20">A37</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c14">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E7:8_1_6"><i><font color="Green" title="E14">A31</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/>
<a NAME="E22:8_1_6"/><i><font color="Green" title="E21">A38</font></i>: 
( <font color="Maroon" title="c21">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c14">c'</font> &amp; <font color="Maroon" title="c20">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E13:8_1"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="yellow18.html#E17:8_1_6"><i><font color="Green" title="E19">A36</font></i></a></i>;<br/>
<i><font color="Green" title="E22">A39</font></i>: <font color="Maroon" title="c22">h</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font>,<font color="Maroon" title="c11">c</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c20">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E11:8_1_6"><i><font color="Green" title="E15">A32</font></i></a>, <a class="txt" href="yellow18.html#E13:8_1_6"><i><font color="Green" title="E16">A33</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a>, <a class="txt" href="yellow18.html#E17:8_1_6"><i><font color="Green" title="E19">A36</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c21">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c20">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E17:8_1_6"><i><font color="Green" title="E19">A36</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c14">c'</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">g</font>,<font color="Maroon" title="c20">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E14:8_1"><i><font color="Green" title="E13">A27</font></i></a>, <a class="txt" href="yellow18.html#E17:8_1_6"><i><font color="Green" title="E19">A36</font></i></a></i>
;<br/>
<a NAME="E24:8_1_6"/>
<font color="Maroon" title="c22">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c14">c'</font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E13:8_1"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="yellow18.html#E15:8_1_6"><i><font color="Green" title="E18">A35</font></i></a></i>;<br/>
<a NAME="E25:8_1_6"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c14">c'</font></span>)</span> <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"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c14">c'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E26:8_1_6"/><b>then </b>
( <font color="Maroon" title="c16">gf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c14">c'</font></span>)</span> &amp; <font color="Maroon" title="c22">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c12">a'</font>,<font color="Maroon" title="c13">b'</font>,<font color="Maroon" title="c14">c'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">gf</font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E17:8_1_6"><i><font color="Green" title="E19">A36</font></i></a>, <a class="txt" href="yellow18.html#E22:8_1_6"><i><font color="Green" title="E21">A38</font></i></a>, <a class="txt" href="yellow18.html#E23:8_1_6"><i><font color="Green" title="E22">A39</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E27:8_1_6"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E13:8_1_6"><i><font color="Green" title="E16">A33</font></i></a>, <a class="txt" href="yellow18.html#E21:8_1_6"><i><font color="Green" title="E20">A37</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">C</font> = <font color="Maroon" title="c4">C</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E17:8_1"/>
for <font color="Olive" title="b1">o</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">C</font><br/> for <font color="Olive" title="b2">o'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o'</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b2">o'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Olive" title="b1">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c5">C</font>;<br/><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>



<b>assume </b><a NAME="E1:8_1_7"/><i><font color="Green" title="E14">A40</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 ;<br/>

<a NAME="E2:8_1_7"/><b>then </b>
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c7">b</font>] )
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="yellow18.html#E12:8_1"><i><font color="Green" title="E11">A25</font></i></a></i>;<br/>
<a NAME="E3:8_1_7"/><b>then </b>
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c7">b</font>, <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">b</font>] )
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:8"><i><font color="Green" title="E3">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:8_1_7"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E13:8_1"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="yellow18.html#E1:8_1_7"><i><font color="Green" title="E14">A40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">C</font> = <font color="Maroon" title="c5">C</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="altcat_2.html#D14" title="ALTCAT_2:def.14">ALTCAT_2:def 14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">C</font>
;<br/>

<b>thus </b><a NAME="E19:8_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">a</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c6">C</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:8_1"><i><font color="Green" title="E4">A4</font></i></a>, <a class="txt" href="yellow18.html#E12:8_1"><i><font color="Green" title="E11">A25</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c6">C</font>;<br/>



<b>assume </b><a NAME="E20:8_1"/><i><font color="Green" title="E14">A41</font></i>: 
( <font color="Maroon" title="c9">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> &amp; <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">b</font> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c11">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>;<br/>

<b>thus </b><a NAME="E21:8_1"/>
( <font color="Maroon" title="c11">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c11">f</font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E13:8_1"><i><font color="Green" title="E12">A26</font></i></a>, <a class="txt" href="yellow18.html#E20:8_1"><i><font color="Green" title="E14">A41</font></i></a></i>;<br/>


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