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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(),  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(),  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of $1,$2) <span class="kw">-&gt; </span>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> $1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> $2</span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> $3;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()) <span class="kw">-&gt; </span>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>2</sub></font>()) = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> $1;<br/>
<a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
( <font color="Maroon">F<sub>3</sub></font>() is  <a href="functor0.html#V19" title="FUNCTOR0:attr.19">injective</a>  &amp; <font color="Maroon">F<sub>3</sub></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="#E1:56"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="ref" href="functor0.html#D36" title="FUNCTOR0:def.36">FUNCTOR0:def 36</a></span>;<br/>
<a NAME="E2:56_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
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">F<sub>5</sub></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>  holds <br/>for <font color="Olive" title="b3">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>  ex <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>() ex <font color="Olive" title="b6">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b4">c</font>) &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b5">d</font>) &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">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>  &amp; <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">d</font>,<font color="Olive" title="b6">g</font>) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:56_1_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>5</sub></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>2</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D11" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>  ex <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>() ex <font color="Olive" title="b4">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">d</font> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">c</font>) &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">d</font>) &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">c</font>,<font color="Olive" title="b3">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>  &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">d</font>,<font color="Olive" title="b4">g</font>) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:56_1_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>  ex <font color="Olive" title="b2">c</font>, <font color="Olive" title="b3">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>() ex <font color="Olive" title="b4">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">d</font> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">c</font>) &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b3">d</font>) &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">c</font>,<font color="Olive" title="b3">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>  &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">c</font>,<font color="Olive" title="b3">d</font>,<font color="Olive" title="b4">g</font>) )</span><br/>

<a NAME="E3:56_1_1"/>
( <font color="Maroon" title="c1">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">F<sub>5</sub></font>() &amp; <font color="Maroon" title="c2">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">F<sub>5</sub></font>() )
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a1</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">b1</font> = <font color="Maroon" title="c2">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>() ex <font color="Olive" title="b3">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">c</font>) &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">d</font>) &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">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>  &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">g</font>) )</span><br/>

<a NAME="E5:56_1_1"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c5">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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f1</font> = <font color="Maroon" title="c5">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">c1</font>, <font color="Maroon" title="c8">d1</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c9">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">c1</font>,<font color="Maroon" title="c8">d1</font><span class="kw"> such that </span><br/><a NAME="E8:56_1_1"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
( <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">c1</font> &amp; <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">d1</font> )
 <span class="kw">and </span><br/><a NAME="E9:56_1_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">c1</font>,<font color="Maroon" title="c8">d1</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> 
 <span class="kw">and </span><br/><a NAME="E10:56_1_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c9">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E5:56_1_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="yellow20.html#T34" target="_self" title="YELLOW20:th.34">Th34</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c7">c1</font>, <font color="Maroon" title="c11">d</font> = <font color="Maroon" title="c8">d1</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="txt" href="#E8:56_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c9">g1</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:56_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E8:56_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E9:56_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E10:56_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">d</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>() ex <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">c</font>,<font color="Olive" title="b1">d</font> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">c</font>) &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">d</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="c10">c</font>,<font color="Olive" title="b1">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>  &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">g</font>) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font> st <br/>( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">c</font>) &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">d</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="c10">c</font>,<font color="Maroon" title="c11">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>  &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Olive" title="b1">g</font>) )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c12">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">c</font>) &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">d</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="c10">c</font>,<font color="Maroon" title="c11">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>  &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">g</font>) )</span><br/>

<a NAME="E13:56_1_1"/>
<font color="Maroon" title="c9">g1</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="c10">c</font>,<font color="Maroon" title="c11">d</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E2:56_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E8:56_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E9:56_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="txt" href="#E10:56_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:56_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c10">c</font>) &amp; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">d</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="c10">c</font>,<font color="Maroon" title="c11">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>  &amp; <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c10">c</font>,<font color="Maroon" title="c11">d</font>,<font color="Maroon" title="c12">g</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E10:56_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:56_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>4</sub></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>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#D11" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E37">A13</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()</span><br/><a NAME="E1:56_1_2"/>
<font color="Maroon" title="c1">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">F<sub>4</sub></font>()
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">b</font> = <font color="Maroon" title="c1">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>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a></span>;<br/><a NAME="E3:56_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>;<br/><span class="kw">hence </span><a NAME="E4:56_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E32">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E38">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">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>5</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:56_1_3"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">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>5</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">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>5</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>)</span><br/><a NAME="E2:56_1_3"/>
( <font color="Maroon" title="c1">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">F<sub>4</sub></font>() &amp; <font color="Maroon" title="c2">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">F<sub>4</sub></font>() )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">c</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c5">d</font> = <font color="Maroon" title="c2">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a></span>;<br/><a NAME="E4:56_1_3"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c3">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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_3"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c3">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a9</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c8">b9</font> = <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a></span>;<br/><a NAME="E7:56_1_3"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c4">c</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>;<br/><a NAME="E8:56_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_3"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>;<br/><a NAME="E9:56_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c3">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="c7">a9</font>,<font color="Maroon" title="c8">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E4:56_1_3"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E7:56_1_3"><span class="lab"><font color="Green" title="E41">A17</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E10:56_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">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>5</sub></font>() <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E6:56_1"/>
( <font color="Maroon">F<sub>4</sub></font>() is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>5</sub></font>() is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon">F<sub>2</sub></font>() )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow20.html#D4" target="_self" title="YELLOW20:def.4">YELLOW20:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>() st <br/>( <font color="Olive" title="b1">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; ( for <font color="Olive" title="b2">a9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()<br/> for <font color="Olive" title="b3">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>()  st <font color="Olive" title="b2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> holds <br/><font color="Olive" title="b1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b3">a</font> ) &amp; ( for <font color="Olive" title="b2">b9</font>, <font color="Olive" title="b3">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()<br/> for <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</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 <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b9</font>,<font color="Olive" title="b3">c9</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; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b3">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">c</font> holds <br/>for <font color="Olive" title="b6">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b9</font>,<font color="Olive" title="b3">c9</font><br/> for <font color="Olive" title="b7">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font>  st <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f</font> holds <br/><font color="Olive" title="b1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">f</font> ) )</span><br/>

<a NAME="E7:56_1"/><span class="lab"><font color="Green" title="E39">A18</font></span>: 
( <font color="Maroon">F<sub>3</sub></font>() is  <a href="functor0.html#V4" title="FUNCTOR0:attr.4">one-to-one</a>  &amp; <font color="Maroon">F<sub>3</sub></font>() is  <a href="functor0.html#V17" title="FUNCTOR0:attr.17">faithful</a>  &amp; <font color="Maroon">F<sub>3</sub></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="#E1:56_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="functor0.html#D34" title="FUNCTOR0:def.34">FUNCTOR0:def 34</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E40">A19</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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>  implies for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>  st <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b2">g</font>) holds <br/><font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> )</span><br/><span class="kw">assume </span><a NAME="E1:56_1_4"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>  st <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b2">g</font>) holds <br/><font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font></span><br/><a NAME="E2:56_1_4"/>
( <font color="Maroon" title="c1">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">F<sub>4</sub></font>() &amp; <font color="Maroon" title="c2">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">F<sub>4</sub></font>() )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a1</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">b1</font> = <font color="Maroon" title="c2">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c5">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c6">g</font>) implies <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> )</span><br/><a NAME="E4:56_1_4"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c5">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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_4"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></span>;<br/><a NAME="E5:56_1_4"/>
<font color="Maroon" title="c6">g</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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_4"><span class="lab"><font color="Green" title="E41">A20</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c8">g1</font> = <font color="Maroon" title="c6">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">b1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56_1_4"><span class="lab"><font color="Green" title="E42">A21</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E7:56_1_4"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c5">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c6">g</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/><span class="kw">then </span><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_4"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">g1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_4"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>
;<br/><span class="kw">hence </span><a NAME="E9:56_1_4"/>
<font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="txt" href="#E4:56_1_4"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="yellow20.html#T33" target="_self" title="YELLOW20:th.33">Th33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E41">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">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>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><br/> for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font>, <font color="Olive" title="b5">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>) &amp; <font color="Olive" title="b5">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>) holds <br/>for <font color="Olive" title="b6">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font><br/> for <font color="Olive" title="b7">g9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b5">c9</font>  st <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>) &amp; <font color="Olive" title="b7">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">g</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b6">f9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:56_1_5"/><span class="lab"><font color="Green" title="E42">A23</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">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> 
 <span class="kw">and </span><br/><a NAME="E2:56_1_5"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font><br/> for <font color="Olive" title="b2">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><br/> for <font color="Olive" title="b3">a9</font>, <font color="Olive" title="b4">b9</font>, <font color="Olive" title="b5">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()  st <font color="Olive" title="b3">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) &amp; <font color="Olive" title="b4">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>) &amp; <font color="Olive" title="b5">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>) holds <br/>for <font color="Olive" title="b6">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">a9</font>,<font color="Olive" title="b4">b9</font><br/> for <font color="Olive" title="b7">g9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b9</font>,<font color="Olive" title="b5">c9</font>  st <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>) &amp; <font color="Olive" title="b7">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">g</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b6">f9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><br/> for <font color="Olive" title="b2">a9</font>, <font color="Olive" title="b3">b9</font>, <font color="Olive" title="b4">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()  st <font color="Olive" title="b2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) &amp; <font color="Olive" title="b3">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>) &amp; <font color="Olive" title="b4">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>) holds <br/>for <font color="Olive" title="b5">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">a9</font>,<font color="Olive" title="b3">b9</font><br/> for <font color="Olive" title="b6">g9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b9</font>,<font color="Olive" title="b4">c9</font>  st <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">f</font>) &amp; <font color="Olive" title="b6">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">g</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b5">f9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a9</font>, <font color="Olive" title="b2">b9</font>, <font color="Olive" title="b3">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>) &amp; <font color="Olive" title="b3">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>) holds <br/>for <font color="Olive" title="b4">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a9</font>,<font color="Olive" title="b2">b9</font><br/> for <font color="Olive" title="b5">g9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b9</font>,<font color="Olive" title="b3">c9</font>  st <font color="Olive" title="b4">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">f</font>) &amp; <font color="Olive" title="b5">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">f9</font></span><br/><a NAME="E3:56_1_5"/><span class="lab"><font color="Green" title="E44">A25</font></span>: 
<font color="Maroon" title="c3">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">F<sub>4</sub></font>()
 ;<br/><a NAME="E4:56_1_5"/>
( <font color="Maroon" title="c1">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">F<sub>4</sub></font>() &amp; <font color="Maroon" title="c2">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">F<sub>4</sub></font>() )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">a1</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c7">b1</font> = <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c8">c1</font> = <font color="Maroon" title="c3">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>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E3:56_1_5"><span class="lab"><font color="Green" title="E44">A25</font></span></a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c9">a9</font>, <font color="Maroon" title="c10">b9</font>, <font color="Maroon" title="c11">c9</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) &amp; <font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>) implies for <font color="Olive" title="b1">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font><br/> for <font color="Olive" title="b2">g9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>  st <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">f</font>) &amp; <font color="Olive" title="b2">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">f9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:56_1_5"/><span class="lab"><font color="Green" title="E45">A26</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>)
 <span class="kw">and </span><br/><a NAME="E7:56_1_5"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>)
 <span class="kw">and </span><br/><a NAME="E8:56_1_5"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font><br/> for <font color="Olive" title="b2">g9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>  st <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">f</font>) &amp; <font color="Olive" title="b2">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b1">f9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">f9</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a9</font>,<font color="Maroon" title="c10">b9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>  st <font color="Maroon" title="c12">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">f</font>) &amp; <font color="Olive" title="b1">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c13">g9</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b9</font>,<font color="Maroon" title="c11">c9</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">f</font>) &amp; <font color="Maroon" title="c13">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font>) implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f9</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:56_1_5"/><span class="lab"><font color="Green" title="E48">A29</font></span>: 
<font color="Maroon" title="c12">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">f</font>)
 <span class="kw">and </span><br/><a NAME="E10:56_1_5"/><span class="lab"><font color="Green" title="E49">A30</font></span>: 
<font color="Maroon" title="c13">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f9</font></span><br/><a NAME="E11:56_1_5"/><span class="lab"><font color="Green" title="E50">A31</font></span>: 
<font color="Maroon" title="c10">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c7">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_1_5"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>;<br/><a NAME="E12:56_1_5"/><span class="lab"><font color="Green" title="E51">A32</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">b1</font>,<font color="Maroon" title="c8">c1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c5">g</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="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_5"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">g1</font> = <font color="Maroon" title="c5">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font> ;<br/><a NAME="E14:56_1_5"/><span class="lab"><font color="Green" title="E52">A33</font></span>: 
<font color="Maroon" title="c11">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56_1_5"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>;<br/><a NAME="E15:56_1_5"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">a1</font>,<font color="Maroon" title="c7">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c4">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="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_5"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">f1</font> = <font color="Maroon" title="c4">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c7">b1</font> ;<br/><a NAME="E17:56_1_5"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
<font color="Maroon" title="c9">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c6">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:56_1_5"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>;<br/><a NAME="E18:56_1_5"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
<font color="Maroon" title="c13">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c14">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56_1_5"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="txt" href="#E10:56_1_5"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>;<br/><a NAME="E19:56_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A37</font></span>: 
<font color="Maroon" title="c13">g9</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="c10">b9</font>,<font color="Maroon" title="c11">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E12:56_1_5"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E11:56_1_5"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="txt" href="#E14:56_1_5"><span class="lab"><font color="Green" title="E52">A33</font></span></a></span>;<br/><a NAME="E20:56_1_5"/><span class="lab"><font color="Green" title="E57">A38</font></span>: 
<font color="Maroon" title="c12">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c15">f1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_5"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E9:56_1_5"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>;<br/><a NAME="E21:56_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A39</font></span>: 
<font color="Maroon" title="c12">f9</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">a9</font>,<font color="Maroon" title="c10">b9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="txt" href="#E15:56_1_5"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E17:56_1_5"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E11:56_1_5"><span class="lab"><font color="Green" title="E50">A31</font></span></a></span>;<br/><a NAME="E22:56_1_5"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">f1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_5"><span class="lab"><font color="Green" title="E42">A23</font></span></a>, <a class="txt" href="#E2:56_1_5"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></span>;<br/><a NAME="E23:56_1_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c15">f1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>;<br/><span class="kw">hence </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">f</font>) = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c14">g1</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c15">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:56_1_5"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E12:56_1_5"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="ref" href="functor0.html#D24" title="FUNCTOR0:def.24">FUNCTOR0:def 24</a></span>
<br/>.= 
<font color="Maroon" title="c13">g9</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56_1_5"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E11:56_1_5"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="txt" href="#E14:56_1_5"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="txt" href="#E20:56_1_5"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E18:56_1_5"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E21:56_1_5"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E19:56_1_5"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E42">A40</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>, <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">a9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a9</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>) implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>, <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a9</font> )</span><br/><span class="kw">assume </span><a NAME="E1:56_1_6"/><span class="lab"><font color="Green" title="E43">A41</font></span>: 
<font color="Maroon" title="c2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>, <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a9</font></span><br/><a NAME="E2:56_1_6"/>
<font color="Maroon" title="c1">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">F<sub>4</sub></font>()
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a1</font> = <font color="Maroon" title="c1">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>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>, <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c1">a</font>) = 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">a1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor2.html#T2" title="FUNCTOR2:th.2">FUNCTOR2:2</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_6"><span class="lab"><font color="Green" title="E43">A41</font></span></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a>, <a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c1">G</font> being   <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a>   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>4</sub></font>(),<font color="Maroon">F<sub>5</sub></font>()<span class="kw"> such that </span><br/><a NAME="E12:56_1"/><span class="lab"><font color="Green" title="E43">A42</font></span>: 
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>4</sub></font>() holds  <font color="Maroon" title="c1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a</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>)
 <span class="kw">and </span><br/><a NAME="E13:56_1"/><span class="lab"><font color="Green" title="E44">A43</font></span>: 
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">F<sub>4</sub></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>  holds <br/>for <font color="Olive" title="b3">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> holds  <font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="yellow18.html#S8" title="YELLOW18:sch.8">YELLOW18:sch 8</a>(<a class="txt" href="#E4:56_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E5:56_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E9:56_1"><span class="lab"><font color="Green" title="E41">A22</font></span></a>, <a class="txt" href="#E10:56_1"><span class="lab"><font color="Green" title="E42">A40</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c1">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; ( for <font color="Olive" title="b1">a9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()<br/> for <font color="Olive" title="b2">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>()  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font> ) &amp; ( for <font color="Olive" title="b1">b9</font>, <font color="Olive" title="b2">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()<br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</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 <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</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; <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E45">A44</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">b</font>) implies <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )</span><br/><a NAME="E1:56_1_7"/>
( <font color="Maroon" title="c2">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">F<sub>4</sub></font>() &amp; <font color="Maroon" title="c3">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">F<sub>4</sub></font>() )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">a1</font> = <font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c5">b1</font> = <font color="Maroon" title="c3">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:56_1_7"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c3">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font></span><br/><span class="kw">then </span><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c4">a1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">b1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:56_1_7"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:56_1"><span class="lab"><font color="Green" title="E39">A18</font></span></a>, <a class="ref" href="yellow20.html#T32" target="_self" title="YELLOW20:th.32">Th32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E15:56_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> 
 <span class="kw">from</span> <span class="lab"><a class="ref" href="yellow18.html#S10" title="YELLOW18:sch.10">YELLOW18:sch 10</a>(<a class="txt" href="#E12:56_1"><span class="lab"><font color="Green" title="E43">A42</font></span></a>, <a class="txt" href="#E13:56_1"><span class="lab"><font color="Green" title="E44">A43</font></span></a>, <a class="txt" href="#E14:56_1"><span class="lab"><font color="Green" title="E45">A44</font></span></a>, <a class="txt" href="#E8:56_1"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="txt" href="#E2:56_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">a9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()<br/> for <font color="Olive" title="b2">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>()  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font> ) &amp; ( for <font color="Olive" title="b1">b9</font>, <font color="Olive" title="b2">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()<br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</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 <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</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; <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font> ) )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b9</font>, <font color="Olive" title="b2">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>()<br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</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 <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</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; <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a1</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="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a1</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a1</font> implies <font color="Maroon" title="c1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">a1</font> )</span><br/><span class="kw">assume </span><a NAME="E1:56_1_8"/><span class="lab"><font color="Green" title="E46">A45</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">a1</font></span><br/><span class="kw">thus </span><font color="Maroon" title="c1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:56_1"><span class="lab"><font color="Green" title="E43">A42</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c3">a1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56_1_8"><span class="lab"><font color="Green" title="E46">A45</font></span></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>4</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">b</font>, <font color="Olive" title="b2">c</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 <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">c</font> holds <br/>for <font color="Olive" title="b3">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><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">b</font>,<font color="Olive" title="b2">c</font>  st <font color="Olive" title="b3">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b3">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Olive" title="b1">b</font>,<font color="Olive" title="b2">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">b1</font>, <font color="Maroon" title="c5">c1</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b1</font> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font> implies for <font color="Olive" title="b1">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font>  st <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E17:56_1"/><span class="lab"><font color="Green" title="E46">A46</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">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> 
 <span class="kw">and </span><br/><a NAME="E18:56_1"/><span class="lab"><font color="Green" title="E47">A47</font></span>: 
( <font color="Maroon" title="c4">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> &amp; <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font>  st <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font>  st <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">f1</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> implies <font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f1</font> )</span><br/>



<span class="kw">assume </span><a NAME="E19:56_1"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f1</font></span><br/>

<a NAME="E20:56_1"/><span class="lab"><font color="Green" title="E49">A49</font></span>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c6">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="c2">b</font>,<font color="Maroon" title="c3">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56_1"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="txt" href="#E18:56_1"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></span>;<br/>
<a NAME="E21:56_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A50</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c4">b1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c5">c1</font></span>)</span></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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">FUNCTOR0:def 19</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="yellow20.html#K4" title="YELLOW20:func.4">|</a> <font color="Maroon">F<sub>4</sub></font>()</span>)</span> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c6">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56_1"><span class="lab"><font color="Green" title="E44">A43</font></span></a>, <a class="txt" href="#E17:56_1"><span class="lab"><font color="Green" title="E46">A46</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56_1"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="txt" href="#E18:56_1"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E19:56_1"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="functor0.html#K5" title="FUNCTOR0:func.5">Morph-Map</a> <font color="Maroon">F<sub>3</sub></font>(),<font color="Maroon" title="c4">b1</font>,<font color="Maroon" title="c5">c1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:56_1"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E21:56_1"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="ref" href="functor0.html#D16" title="FUNCTOR0:def.16">FUNCTOR0:def 16</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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