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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>  &amp;  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a>  &amp;  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">G</font> = the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">C</font> = the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a>  &amp;  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">o1</font>, <font color="Maroon" title="c5">o2</font>, <font color="Maroon" title="c6">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</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="c5">o2</font>,<font color="Maroon" title="c6">o3</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 <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</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="c5">o2</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:43_1"/><span class="kw">then </span>
(  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c5">o2</font> <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;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c5">o2</font>,<font color="Maroon" title="c6">o3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>;<br/>
<a NAME="E3:43_1"/><span class="kw">then </span>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#T5" target="_self" title="ALTCAT_1:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:43_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o1</font>,<font color="Maroon" title="c6">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:43"/>
 <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font> is  <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c7">l</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_1.html#D7" target="_self" title="ALTCAT_1:def.7">ALTCAT_1:def 7</a>,<a class="ref" href="altcat_1.html#D17" target="_self" title="ALTCAT_1:def.17">ALTCAT_1:def 17</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">g</font>,<font color="Olive" title="b1">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">g</font></span>)</span>,<font color="Olive" title="b1">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">i9</font> = <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c9">j9</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c10">k9</font> = <font color="Maroon" title="c6">k</font>, <font color="Maroon" title="c11">l9</font> = <font color="Maroon" title="c7">l</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">f</font>, <font color="Maroon" title="c13">g</font>, <font color="Maroon" title="c14">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c14">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font></span>)</span>,<font color="Maroon" title="c12">f</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_2"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c12">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
 <span class="kw">and </span><br/><a NAME="E3:43_2"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c13">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E4:43_2"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c14">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font></span>)</span>,<font color="Maroon" title="c12">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c15">h99</font> = <font color="Maroon" title="c14">h</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">k9</font>,<font color="Maroon" title="c11">l9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">g99</font> = <font color="Maroon" title="c13">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">j9</font>,<font color="Maroon" title="c10">k9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<a NAME="E7:43_2"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">k9</font>,<font color="Maroon" title="c11">l9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>;<br/>
<a NAME="E8:43_2"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i9</font>,<font color="Maroon" title="c9">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</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="c9">j9</font>,<font color="Maroon" title="c10">k9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">f9</font> = <font color="Maroon" title="c12">f</font>, <font color="Maroon" title="c18">g9</font> = <font color="Maroon" title="c13">g</font>, <font color="Maroon" title="c19">h9</font> = <font color="Maroon" title="c14">h</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a></span>;<br/>
<a NAME="E10:43_2"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c10">k9</font>,<font color="Maroon" title="c11">l9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E11:43_2"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j9</font>,<font color="Maroon" title="c10">k9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<a NAME="E12:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j9</font>,<font color="Maroon" title="c11">l9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E10:43_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></span>;<br/>
<a NAME="E13:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c15">h99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">g99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c18">g9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">f99</font> = <font color="Maroon" title="c12">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">i9</font>,<font color="Maroon" title="c9">j9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a></span>;<br/>
<a NAME="E15:43_2"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i9</font>,<font color="Maroon" title="c9">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
;<br/>
<a NAME="E16:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i9</font>,<font color="Maroon" title="c10">k9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E11:43_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></span>;<br/>
<a NAME="E17:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c16">g99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c20">f99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">f9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></span>;<br/>
<a NAME="E18:43_2"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">i9</font>,<font color="Maroon" title="c11">l9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E10:43_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E16:43_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></span>;<br/>
<a NAME="E19:43_2"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">g99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></span>;<br/>
<a NAME="E20:43_2"/>
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">g99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c20">f99</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">g</font>,<font color="Maroon" title="c12">f</font></span>)</span> = 
<font color="Maroon" title="c15">h99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">g99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c20">f99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E16:43_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></span>
<br/>.= 
<font color="Maroon" title="c19">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">f9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E16:43_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E18:43_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E17:43_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">h9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c18">g9</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c17">f9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">h99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c16">g99</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c20">f99</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E12:43_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E18:43_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E13:43_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p3">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">l</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">h</font>,<font color="Maroon" title="c13">g</font></span>)</span>,<font color="Maroon" title="c12">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E12:43_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E19:43_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:43"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is  <a href="altcat_1.html#V5" title="ALTCAT_1:attr.5">with_left_units</a> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_1.html#D18" target="_self" title="ALTCAT_1:def.18">ALTCAT_1:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> is  <a href="altcat_1.html#V4" title="ALTCAT_1:attr.4">with_right_units</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_1.html#D9" target="_self" title="ALTCAT_1:def.9">ALTCAT_1:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b2">i</font> being   <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">e</font>,<font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">i9</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<span class="kw">take </span>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<a NAME="E2:43_3"/><span class="lab"><font color="Green" title="E31">A14</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">i9</font>,<font color="Maroon" title="c5">i9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:43_3"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">I</font> =  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">i9</font>,<font color="Maroon" title="c5">i9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_3"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c9">j9</font> = <font color="Maroon" title="c7">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E6:43_3"/><span class="lab"><font color="Green" title="E32">A15</font></span>: 
<font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f99</font> = <font color="Maroon" title="c8">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">j9</font>,<font color="Maroon" title="c5">i9</font> ;<br/>
<a NAME="E8:43_3"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">j9</font>,<font color="Maroon" title="c5">i9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f9</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_3"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">j</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c8">f</font> = 
<font color="Maroon" title="c6">I</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">f99</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_3"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E6:43_3"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c11">f9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_3"><span class="lab"><font color="Green" title="E31">A14</font></span></a>, <a class="txt" href="#E6:43_3"><span class="lab"><font color="Green" title="E32">A15</font></span></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_1.html#D8" target="_self" title="ALTCAT_1:def.8">ALTCAT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b2">j</font> being   <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b3">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">i9</font> = <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<span class="kw">take </span>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<a NAME="E5:43"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">i9</font>,<font color="Maroon" title="c5">i9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:43"/>
 <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">j</font> being   <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Olive" title="b1">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">I</font> =  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">i9</font>,<font color="Maroon" title="c5">i9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="altcat_1.html#T2" target="_self" title="ALTCAT_1:th.2">Th2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c9">j9</font> = <font color="Maroon" title="c7">j</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> ;<br/>
<span class="kw">assume </span><a NAME="E9:43"/><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c8">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 <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f99</font> = <font color="Maroon" title="c8">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">i9</font>,<font color="Maroon" title="c9">j9</font> ;<br/>
<a NAME="E11:43"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">i9</font>,<font color="Maroon" title="c9">j9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" target="_self" title="ALTCAT_1:def.16">Def16</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">f9</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <span class="p2">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">f</font>,<span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span> = 
<font color="Maroon" title="c10">f99</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">I</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></span>
<br/>.= 
<font color="Maroon" title="c11">f9</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <font color="Maroon" title="c4">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="ref" href="altcat_1.html#T18" target="_self" title="ALTCAT_1:th.18">Th18</a></span>
<br/>.= 
<font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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