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<b>thus </b><a NAME="E1:56_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>() )
 ; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">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>()<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">a'</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">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="b3">a</font> ) &amp; ( for <font color="Olive" title="b2">b'</font>, <font color="Olive" title="b3">c'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">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">b'</font>,<font color="Olive" title="b3">c'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b3">c'</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">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b'</font>,<font color="Olive" title="b3">c'</font><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">f'</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">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="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="E2:56_1"/><i><font color="Green" title="E35">A4</font></i>: 
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>()
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a></i>;<br/>
<b>deffunc </b><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>()) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of 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/>
<b>deffunc </b><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>(),  <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) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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/>
<div><i><font color="Green" title="E36">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</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_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>4</sub></font>()
 ;<br/><b>then reconsider </b><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>() <b>by </b><i><a class="txt" href="#E2:56_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/><a NAME="E3:56_1_1"/>
<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>
 <b>by </b><i><a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E4:56_1_1"/>
<font color="Maroon">H<sub>1</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>()
 <b>by </b><i><a class="txt" href="#E2:56"><i><font color="Green" title="E33">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E37">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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>2</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>1</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>) )</span><br/><b>assume </b><a NAME="E1:56_1_2"/><i><font color="Green" title="E38">A7</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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:()"><i><font color="Red">::  thesis: </font></i></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>2</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>1</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>)</span><br/><a NAME="E2:56_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>() &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/><b>then reconsider </b><font color="Maroon" title="c3">c</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c4">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>() <b>by </b><i><a class="txt" href="#E2:56_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">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="c1">a</font>, <font color="Maroon" title="c6">b'</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>() <b>by </b><i><a class="txt" href="#E3:56_1"><i><font color="Green" title="E36">A5</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">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>1</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>)</span><br/><a NAME="E5:56_1_2"/><i><font color="Green" title="E39">A8</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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">c</font>,<font color="Maroon" title="c4">d</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c7">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> )
 <b>by </b><i><a class="txt" href="#E1:56_1_2"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font> ;<br/><a NAME="E7:56_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="c3">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="c4">d</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#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c7">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="c8">g</font> )
 <b>by </b><i><a class="txt" href="#E1:56_1_2"><i><font color="Green" title="E38">A7</font></i></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></i>;<br/><a NAME="E8:56_1_2"/><b>then </b>
<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="c7">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="c5">a'</font>,<font color="Maroon" title="c6">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="#E5:56_1_2"><i><font color="Green" title="E39">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E9:56_1_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">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>1</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E38">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_3"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <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>5</sub></font>()  st <font color="Olive" title="b3">a'</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>) &amp; <font color="Olive" title="b4">b'</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>) &amp; <font color="Olive" title="b5">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>) holds <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">a'</font>,<font color="Olive" title="b4">b'</font><br/> for <font color="Olive" title="b7">g'</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">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>) &amp; <font color="Olive" title="b7">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">g</font>) holds <br/><font color="Maroon">H<sub>2</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">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b6">f'</font> )</span><br/><b>assume </b><a NAME="E1:56_1_3"/><i><font color="Green" title="E39">A10</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <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>5</sub></font>()  st <font color="Olive" title="b3">a'</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>) &amp; <font color="Olive" title="b4">b'</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>) &amp; <font color="Olive" title="b5">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>) holds <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">a'</font>,<font color="Olive" title="b4">b'</font><br/> for <font color="Olive" title="b7">g'</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">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>) &amp; <font color="Olive" title="b7">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b2">g</font>) holds <br/><font color="Maroon">H<sub>2</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">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b6">f'</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>() &amp; <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/><b>then reconsider </b><font color="Maroon" title="c4">a1</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c5">b1</font> = <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c6">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>() <b>by </b><i><a class="txt" href="#E2:56_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c7">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:()"><i><font color="Red">::  thesis: </font></i></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">a'</font>, <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>5</sub></font>()  st <font color="Olive" title="b2">a'</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>) &amp; <font color="Olive" title="b3">b'</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>) &amp; <font color="Olive" title="b4">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>) holds <br/>for <font color="Olive" title="b5">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b3">b'</font><br/> for <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="b3">b'</font>,<font color="Olive" title="b4">c'</font>  st <font color="Olive" title="b5">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">f</font>) &amp; <font color="Olive" title="b6">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">g</font>) holds <br/><font color="Maroon">H<sub>2</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="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b5">f'</font></span><br/><b>let </b><font color="Maroon" title="c8">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">b'</font>, <font color="Olive" title="b3">c'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>5</sub></font>()  st <font color="Olive" title="b1">a'</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>) &amp; <font color="Olive" title="b2">b'</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>) &amp; <font color="Olive" title="b3">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>) holds <br/>for <font color="Olive" title="b4">f'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a'</font>,<font color="Olive" title="b2">b'</font><br/> for <font color="Olive" title="b5">g'</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b'</font>,<font color="Olive" title="b3">c'</font>  st <font color="Olive" title="b4">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">f</font>) &amp; <font color="Olive" title="b5">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font>) holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">f'</font></span><br/><b>let </b><font color="Maroon" title="c9">a'</font>, <font color="Maroon" title="c10">b'</font>, <font color="Maroon" title="c11">c'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">a'</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>) &amp; <font color="Maroon" title="c10">b'</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>) &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>) 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="c9">a'</font>,<font color="Maroon" title="c10">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="c10">b'</font>,<font color="Maroon" title="c11">c'</font>  st <font color="Olive" title="b1">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">f</font>) &amp; <font color="Olive" title="b2">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font>) holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )</span><br/><b>assume </b><a NAME="E4:56_1_3"/><i><font color="Green" title="E40">A11</font></i>: 
( <font color="Maroon" title="c9">a'</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>) &amp; <font color="Maroon" title="c10">b'</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>) &amp; <font color="Maroon" title="c11">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">c</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c9">a'</font>,<font color="Maroon" title="c10">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="c10">b'</font>,<font color="Maroon" title="c11">c'</font>  st <font color="Olive" title="b1">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">f</font>) &amp; <font color="Olive" title="b2">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font>) holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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></span><br/><b>let </b><font color="Maroon" title="c12">f'</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c10">b'</font>,<font color="Maroon" title="c11">c'</font>  st <font color="Maroon" title="c12">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">f</font>) &amp; <font color="Olive" title="b1">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font>) holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f'</font></span><br/><b>let </b><font color="Maroon" title="c13">g'</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c11">c'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c12">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">f</font>) &amp; <font color="Maroon" title="c13">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font>) implies <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f'</font> )</span><br/><b>assume </b><a NAME="E5:56_1_3"/><i><font color="Green" title="E41">A12</font></i>: 
( <font color="Maroon" title="c12">f'</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>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c7">f</font>) &amp; <font color="Maroon" title="c13">g'</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>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f'</font></span><br/><a NAME="E6:56_1_3"/><i><font color="Green" title="E42">A13</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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">a1</font>,<font color="Maroon" title="c5">b1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c7">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> )
 <b>by </b><i><a class="txt" href="#E1:56_1_3"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">f1</font> = <font color="Maroon" title="c7">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">b1</font> ;<br/><a NAME="E8:56_1_3"/><i><font color="Green" title="E43">A14</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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="c5">b1</font>,<font color="Maroon" title="c6">c1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c8">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> )
 <b>by </b><i><a class="txt" href="#E1:56_1_3"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">g1</font> = <font color="Maroon" title="c8">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">b1</font>,<font color="Maroon" title="c6">c1</font> ;<br/><a NAME="E10:56_1_3"/><i><font color="Green" title="E44">A15</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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> 
 <b>by </b><i><a class="txt" href="#E1:56_1_3"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/><a NAME="E11:56_1_3"/><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c14">f1</font>
 <b>by </b><i><a class="txt" href="#E1:56_1_3"><i><font color="Green" title="E39">A10</font></i></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></i>;<br/><a NAME="E12:56_1_3"/><i><font color="Green" title="E46">A17</font></i>: 
( <font color="Maroon" title="c9">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">a1</font> &amp; <font color="Maroon" title="c10">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">b1</font> &amp; <font color="Maroon" title="c11">c'</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">c1</font> )
 <b>by </b><i><a class="txt" href="#E4:56_1_3"><i><font color="Green" title="E40">A11</font></i></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></i>;<br/><a NAME="E13:56_1_3"/><i><font color="Green" title="E47">A18</font></i>: 
( <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="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">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="c15">g1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c14">f1</font></span>)</span> &amp; <font color="Maroon" title="c12">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="c14">f1</font> &amp; <font color="Maroon" title="c13">g'</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">g1</font> )
 <b>by </b><i><a class="txt" href="#E1:56_1_3"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="#E5:56_1_3"><i><font color="Green" title="E41">A12</font></i></a>, <a class="txt" href="#E10:56_1_3"><i><font color="Green" title="E44">A15</font></i></a>, <a class="txt" href="#E11:56_1_3"><i><font color="Green" title="E45">A16</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></i>;<br/><a NAME="E14:56_1_3"/><b>then </b><i><font color="Green" title="E48">A19</font></i>: 
( <font color="Maroon" title="c12">f'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c13">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="c10">b'</font>,<font color="Maroon" title="c11">c'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="#E6:56_1_3"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="#E8:56_1_3"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="#E12:56_1_3"><i><font color="Green" title="E46">A17</font></i></a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">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="c15">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="c14">f1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:56_1_3"><i><font color="Green" title="E42">A13</font></i></a>, <a class="txt" href="#E8:56_1_3"><i><font color="Green" title="E43">A14</font></i></a>, <a class="txt" href="#E13:56_1_3"><i><font color="Green" title="E47">A18</font></i></a>, <a class="ref" href="functor0.html#D24" title="FUNCTOR0:def.24">FUNCTOR0:def 24</a></i>
<br/>.= 
<font color="Maroon" title="c13">g'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">f'</font>
<b>by </b><i><a class="txt" href="#E12:56_1_3"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="#E13:56_1_3"><i><font color="Green" title="E47">A18</font></i></a>, <a class="txt" href="#E14:56_1_3"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E39">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_4"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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>5</sub></font>()  st <font color="Olive" title="b1">a'</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>) holds <br/><font color="Maroon">H<sub>2</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">a'</font></span><br/><b>let </b><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>5</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a'</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>) implies <font color="Maroon">H<sub>2</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">a'</font> )</span><br/><b>assume </b><a NAME="E1:56_1_4"/><i><font color="Green" title="E40">A21</font></i>: 
<font color="Maroon" title="c2">a'</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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>2</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">a'</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>()
 ;<br/><b>then reconsider </b><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>() <b>by </b><i><a class="txt" href="#E2:56_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/><b>thus </b><font color="Maroon">H<sub>2</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>
<b>by </b><i><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></i>
<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>
<b>by </b><i><a class="ref" href="functor2.html#T2" title="FUNCTOR2:th.2">FUNCTOR2:2</a></i>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">a'</font>
<b>by </b><i><a class="txt" href="#E1:56_1_4"><i><font color="Green" title="E40">A21</font></i></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></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><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>()<b> such that </b><br/><a NAME="E8:56_1"/><i><font color="Green" title="E40">A22</font></i>: 
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>1</sub></font>(<font color="Olive" title="b1">a</font>)
 <b>and </b><br/><a NAME="E9:56_1"/><i><font color="Green" title="E41">A23</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">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>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S8" title="YELLOW18:sch.8">YELLOW18:sch 8</a>(<a class="txt" href="#E3:56_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="#E4:56_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="#E5:56_1"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="#E6:56_1"><i><font color="Green" title="E39">A20</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">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>()<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">a'</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">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="b2">a</font> ) &amp; ( 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>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">b'</font>,<font color="Olive" title="b2">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="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c'</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">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><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">f'</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">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="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/>

<a NAME="E10:56_1"/><i><font color="Green" title="E42">A24</font></i>: 
( <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> )
 
<b>by </b><i><a class="txt" href="#E1:56"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="functor0.html#D36" title="FUNCTOR0:def.36">FUNCTOR0:def 36</a></i>;<br/>
<a NAME="E11:56_1"/><b>then </b><i><font color="Green" title="E43">A25</font></i>: 
( <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> )
 
<b>by </b><i><a class="ref" href="functor0.html#D34" title="FUNCTOR0:def.34">FUNCTOR0:def 34</a></i>;<br/>
<div><i><font color="Green" title="E44">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_5"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</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_5"/>
( <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/><b>then reconsider </b><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>() <b>by </b><i><a class="txt" href="#E2:56_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/><b>assume </b><a NAME="E3:56_1_5"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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/><b>then </b><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>
<b>by </b><i><a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></i>
<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>
<b>by </b><i><a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></i>
;<br/><b>hence </b><a NAME="E5:56_1_5"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="txt" href="#E11:56_1"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="yellow20.html#T32" target="_self" title="YELLOW20:th.32">Th32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E45">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_6"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">a</font>,<font color="Maroon" title="c3">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="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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/><b>assume </b><a NAME="E1:56_1_6"/><i><font color="Green" title="E46">A28</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></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="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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_6"/>
( <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/><b>then reconsider </b><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>() <b>by </b><i><a class="txt" href="#E2:56_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c7">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">g</font>) implies <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> )</span><br/><a NAME="E4:56_1_6"/><i><font color="Green" title="E47">A29</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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">a1</font>,<font color="Maroon" title="c5">b1</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">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c7">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">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E1:56_1_6"><i><font color="Green" title="E46">A28</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c6">f</font>, <font color="Maroon" title="c9">g1</font> = <font color="Maroon" title="c7">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">b1</font> ;<br/><b>assume </b><a NAME="E6:56_1_6"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">g</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">g</font></span><br/><b>then </b><font color="Maroon">F<sub>3</sub></font>() <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c8">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="c7">g</font>
<b>by </b><i><a class="txt" href="#E1:56_1_6"><i><font color="Green" title="E46">A28</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></i>
<br/>.= 
<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>
<b>by </b><i><a class="txt" href="#E1:56_1_6"><i><font color="Green" title="E46">A28</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></i>
;<br/><b>hence </b><a NAME="E8:56_1_6"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="txt" href="#E11:56_1"><i><font color="Green" title="E43">A25</font></i></a>, <a class="txt" href="#E4:56_1_6"><i><font color="Green" title="E47">A29</font></i></a>, <a class="ref" href="yellow20.html#T33" target="_self" title="YELLOW20:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E14:56_1"/><i><font color="Green" title="E46">A30</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">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>1</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>1</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>2</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_7"><b>proof </b></a><div class="add">

<b>let </b><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>5</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">a</font>,<font color="Maroon" title="c3">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="c2">a</font>,<font color="Maroon" title="c3">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="c2">a</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>) &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</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>2</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/>

<b>assume </b><a NAME="E1:56_1_7"/><i><font color="Green" title="E47">A31</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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:()"><i><font color="Red">::  thesis: </font></i></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="c2">a</font>,<font color="Maroon" title="c3">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="c2">a</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>) &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</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>2</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="E2:56_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>5</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>5</sub></font>() &amp; 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>() )
 
<b>by </b><i><a class="ref" href="altcat_2.html#D11" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a></i>;<br/>
<b>then reconsider </b><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>2</sub></font>() ;<br/>
<b>let </b><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">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c2">a</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>) &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">g</font>) )</span><br/>

<a NAME="E4:56_1_7"/><i><font color="Green" title="E48">A32</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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">a1</font>,<font color="Maroon" title="c5">b1</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">a</font>,<font color="Maroon" title="c3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="#E1:56_1_7"><i><font color="Green" title="E47">A31</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c6">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">b1</font> ;<br/>
<b>consider </b><font color="Maroon" title="c8">c1</font>, <font color="Maroon" title="c9">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="c10">g1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c9">d1</font><b> such that </b><br/><a NAME="E7:56_1_7"/><i><font color="Green" title="E49">A33</font></i>: 
( <font color="Maroon" title="c4">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="c8">c1</font> &amp; <font color="Maroon" title="c5">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="c9">d1</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="c8">c1</font>,<font color="Maroon" title="c9">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>  &amp; <font color="Maroon" title="c7">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="c10">g1</font> )
 <b>by </b><i><a class="txt" href="#E10:56_1"><i><font color="Green" title="E42">A24</font></i></a>, <a class="txt" href="#E4:56_1_7"><i><font color="Green" title="E48">A32</font></i></a>, <a class="ref" href="yellow20.html#T34" target="_self" title="YELLOW20:th.34">Th34</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">c</font> = <font color="Maroon" title="c8">c1</font>, <font color="Maroon" title="c12">d</font> = <font color="Maroon" title="c9">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>() <b>by </b><i><a class="txt" href="#E2:56"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="#E7:56_1_7"><i><font color="Green" title="E49">A33</font></i></a></i>;<br/>
<a NAME="E9:56_1_7"/><i><font color="Green" title="E50">A34</font></i>: 
<font color="Maroon" title="c10">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="c11">c</font>,<font color="Maroon" title="c12">d</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="#E1:56_1_7"><i><font color="Green" title="E47">A31</font></i></a>, <a class="txt" href="#E7:56_1_7"><i><font color="Green" title="E49">A33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c10">g1</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font> <b>by </b><i><a class="txt" href="#E3:56"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="#E1:56_1_7"><i><font color="Green" title="E47">A31</font></i></a>, <a class="txt" href="#E7:56_1_7"><i><font color="Green" title="E49">A33</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">c</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c11">c</font>,<font color="Olive" title="b1">d</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">c</font>) &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</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="c11">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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">c</font>,<font color="Olive" title="b1">d</font>,<font color="Olive" title="b2">g</font>) )</span><br/>

<b>take </b>
<font color="Maroon" title="c12">d</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c11">c</font>,<font color="Maroon" title="c12">d</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">c</font>) &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">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="c11">c</font>,<font color="Maroon" title="c12">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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font>,<font color="Olive" title="b1">g</font>) )</span><br/>

<b>take </b>
<font color="Maroon" title="c13">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">c</font>) &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">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="c11">c</font>,<font color="Maroon" title="c12">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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c13">g</font>) )</span><br/>

<b>thus </b><a NAME="E11:56_1_7"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">c</font>) &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c12">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="c11">c</font>,<font color="Maroon" title="c12">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="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c11">c</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c13">g</font>) )
 <b>by </b><i><a class="txt" href="#E7:56_1_7"><i><font color="Green" title="E49">A33</font></i></a>, <a class="txt" href="#E9:56_1_7"><i><font color="Green" title="E50">A34</font></i></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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E15:56_1"/>
<font color="Maroon" title="c1">G</font> is <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>
 <b>from </b><i><a class="ref" href="yellow18.html#S10" title="YELLOW18:sch.10">YELLOW18:sch 10</a>(<a class="txt" href="#E8:56_1"><i><font color="Green" title="E40">A22</font></i></a>, <a class="txt" href="#E9:56_1"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="#E12:56_1"><i><font color="Green" title="E44">A26</font></i></a>, <a class="txt" href="#E13:56_1"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="#E14:56_1"><i><font color="Green" title="E46">A30</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( 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>()<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">a'</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">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="b2">a</font> ) &amp; ( 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>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">b'</font>,<font color="Olive" title="b2">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="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c'</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">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><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">f'</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">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="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><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>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">b'</font>,<font color="Olive" title="b2">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="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c'</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">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><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">f'</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">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="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"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>assume </b><a NAME="E1:56_1_8"/><i><font color="Green" title="E47">A35</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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/><b>thus </b><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>
<b>by </b><i><a class="txt" href="#E8:56_1"><i><font color="Green" title="E40">A22</font></i></a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:56_1_8"><i><font color="Green" title="E47">A35</font></i></a>, <a class="ref" href="yellow20.html#T29" target="_self" title="YELLOW20:th.29">Th29</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">f'</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">f'</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">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="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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">f'</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">f'</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">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="b2">f</font> )</span><br/>



<b>assume </b><a NAME="E17:56_1"/><i><font color="Green" title="E47">A36</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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="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:()"><i><font color="Red">::  thesis: </font></i></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="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">f'</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">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="b2">f</font></span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>



<b>assume </b><a NAME="E18:56_1"/><i><font color="Green" title="E48">A37</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E19:56_1"/><i><font color="Green" title="E49">A38</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="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> )
 
<b>by </b><i><a class="txt" href="#E17:56_1"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="altcat_2.html#T32" title="ALTCAT_2:th.32">ALTCAT_2:32</a></i>;<br/>
<a NAME="E20:56_1"/><b>then </b><i><font color="Green" title="E50">A39</font></i>: 
<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> 
 
<b>by </b><i><a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">FUNCTOR0:def 19</a></i>;<br/>
<b>thus </b><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>
<b>by </b><i><a class="txt" href="#E9:56_1"><i><font color="Green" title="E41">A23</font></i></a>, <a class="txt" href="#E17:56_1"><i><font color="Green" title="E47">A36</font></i></a></i>
<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>
<b>by </b><i><a class="txt" href="#E17:56_1"><i><font color="Green" title="E47">A36</font></i></a>, <a class="txt" href="#E18:56_1"><i><font color="Green" title="E48">A37</font></i></a>, <a class="ref" href="yellow20.html#T30" target="_self" title="YELLOW20:th.30">Th30</a></i>
<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>
<b>by </b><i><a class="txt" href="#E19:56_1"><i><font color="Green" title="E49">A38</font></i></a>, <a class="txt" href="#E20:56_1"><i><font color="Green" title="E50">A39</font></i></a>, <a class="ref" href="functor0.html#D16" title="FUNCTOR0:def.16">FUNCTOR0:def 16</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
